Merci Également

Viviane Baladi

Gérard Besson

Étienne Ghys

Bruno Sévennec

Étienne

Ghani

Jean-Claude Bruno

Damien Olivier

Romain Merci

Émeric, Fangzhou, François, Léo, Loïc, Marie, Marielle, Maxime, Michele Anne Arthur

Olga Rémi

Sébastien Romain Samuel

Vincent Valentin

Aurore

Benoît, Camille Benjamin

Élodie

Guillaume

Hélène, Irène, Jonas Marthe Guillaume

Mikaël, Nathanaël, Ophélia, Pierre Quentin Laetitia

et tous les autres. Merci à Magalie et Virginia dont la gentillesse et l'efficacité ne sont plus à prouver. L'ENS Lyon est également un cadre privilégié pour débuter en tant qu'enseignant : merci à toute l'équipe enseignante, aux élèves, et à ceux qui ont préparé les TD avec moi et avec qui j'ai le plaisir d'échanger des idées et des exercices (Alexandre et Alexandre, Daniel, Mohamed et Sylvain).

Mais je dois aussi beaucoup aux membres de l'Institut Fourier à Grenoble, où j'ai passé beaucoup de temps durant la seconde moitié de ma thèse. Merci à Gérard Besson pour son accueil chaleureux, à Pierre Dehornoy pour avoir relu et aidé à améliorer un passage de ma thèse, et à Benoît Kloeckner et Frédéric Faure pour des discussions fructueuses. Merci à Kévin et Simon qui ont partagé mon bureau à Grenoble et l'ont rendu vivant, mais aussi à Alejandro

Introduction (français)

Cette thèse est divisée en trois parties qui peuvent être lues indépendamment. Dans la première, on étudie les théorèmes d'universalité pour les mécanismes (aussi appelés systèmes articulés) dont l'espace ambiant est une surface homogène. Dans la seconde, on étudie un lien entre flots géodésiques et billards, ainsi que la dynamique de certains mécanismes. La troisième porte sur les structures de similitude transverses sur les feuilletages, ainsi que sur le théorème de décomposition de De Rham. Chacune de ces parties contient une introduction propre.

Un mécanisme est un ensemble de tiges rigides reliées par des liaisons pivots. Mathématiquement, on considère un mécanisme comme un graphe muni d'une structure supplémentaire : on associe une longueur à chaque arête (le graphe est dit métrique), et certains sommets sont fixés au plan tandis que d'autres évoluent librement. Une réalisation d'un mécanisme dans le plan est le choix d'une position dans le plan pour chaque sommet, de sorte que les longueurs associées aux arêtes correspondent aux distances dans le plan entre les sommets correspondants. En particulier, on autorise les arêtes à se croiser. Enfin, l'espace de configuration d'un système articulé est l'ensemble de ses réalisations.

Le premier chapitre constitue une introduction à cette notion de mécanisme : on y donne des éléments historiques, et des exemples fondamentaux, qui sont utiles pour les chapitres suivants.

La première partie, qui suit cette introduction, est constituée de quatre chapitres et correspond à la pré-publication [START_REF] Kourganoff | Universality theorems for linkages in homogeneous surfaces[END_REF] : on y étudie des mécanismes dont l'espace ambiant n'est plus le plan, mais diverses variétés riemanniennes. Le chapitre 2 introduit la question de l'universalité des mécanismes : cette notion correspond à l'idée que toute courbe serait tracée par un sommet d'un mécanisme, et que toute variété différentiable serait l'espace de configuration d'un mécanisme. On y présentera des résultats déjà connus qui vont dans ce sens pour les mécanismes dans le plan : d'une part, les courbes que l'on peut tracer sont exactement les courbes semi-algébriques compactes ; et d'autre part, pour toute variété compacte connexe M , il existe 1 un espace de configuration d'un système articulé dont toutes les composantes connexes sont difféomorphes à M . Ce même chapitre contient aussi les énoncés de tous les nouveaux résultats essentiels de cette partie, qui consistent à étendre les théorèmes d'universalité au plan de Minkowski, au plan hyperbolique et enfin à la sphère. Dans chaque cas, les difficultés rencontrées diffèrent, ainsi que les techniques pour les résoudre, mais les résultats obtenus sont très similaires, sauf dans le cas du plan de Minkowski, où l'on s'affranchit de l'exigence de compacité : l'universalité est alors valable dans un sens encore plus large. Les trois derniers chapitres contiennent les démonstrations de ces énoncés, alors que les outils généraux sont donnés CONTENTS dans le chapitre 2.

La seconde partie est constituée de trois chapitres, dont les deux derniers correspondent à la pré-publication [START_REF] Kourganoff | Anosov geodesic flows, billiards and linkages[END_REF]. Dans le chapitre 6, on établit un premier lien entre flots géodésiques sur des variétés à courbure négative et billards dispersifs, en mettant en parallèle les comportements de ces deux systèmes. La similitude entre ces deux systèmes est bien connue depuis les travaux de Sinaï dans les années 1960, mais elle est rarement détaillée dans la littérature. Dans le même ordre d'idée, on donne une condition suffisante pour qu'un flot géodésique sur une surface fermée soit Anosov : il suffit que toutes les solutions de l'équation de Ricatti le long des géodésiques, nulles au temps t = 0, soient supérieures à une même constante m > 0 lorsque t = 1. Ce théorème bien connu a été utilisé à plusieurs reprises dans la littérature, mais sans qu'aucune preuve écrite ne semble disponible : nous l'utiliserons à notre tour dans les résultats qui suivent. Dans le chapitre 7, on présente deux résultats nouveaux concernant le flot géodésique de surfaces dans R 3 euclidien, qui ont subi une forte contraction selon l'un des axes. Toute surface dans R 3 peut être aplatie selon l'axe des z, et la surface aplatie s'approche d'une table de billard dans R 2 . On montre que, sous certaines hypothèses, le flot géodésique de la surface converge localement uniformément vers le flot de billard. De plus, si le billard est dispersif, les propriétés chaotiques du billard remontent au flot géodésique : on montre qu'il est alors Anosov. Enfin, dans le chapitre 8, on donne des généralités sur la dynamique des systèmes articulés, puis on applique le résultat du chapitre 7 à la théorie des systèmes articulés. Ceci permet d'obtenir un nouvel exemple de mécanisme Anosov, comportant cinq tiges. C'est la première fois qu'on exhibe un système articulé Anosov dont les longueurs des arêtes sont données explicitement. Une vidéo de ce mécanisme, due à Jos Leys, est disponible sur ma page web.

La troisième partie n'a pas de lien direct avec les deux autres, si ce n'est l'étude de variétés riemanniennes : elle correspond à la pré-publication [START_REF] Kourganoff | Transverse similarity structures on foliations, and de rham decomposition[END_REF]. On s'intéresse d'abord aux variétés munies de connexions localement métriques, c'est-à-dire de connexions qui sont localement des connexions de Levi-Civita de métriques riemanniennes ; on donne dans ce cadre un analogue du théorème de décomposition de De Rham, qui s'applique habituellement aux variétés riemanniennes. Dans le cas où une telle connexion préserve une structure conforme, on montre que cette décomposition comporte au plus deux facteurs ; de plus, lorsqu'il y a exactement deux facteurs, l'un des deux est l'espace euclidien R q . On répond ainsi à une question posée dans [START_REF] Vladimir | Locally conformally berwald manifolds and compact quotients of reducible manifolds by homotheties[END_REF]. L'étude des connexions localement métriques qui préservent une structure conforme est étroitement liée à celle des "structures de similitude" sur les variétés : ce sont les structures obtenues par quotient d'une variété riemannienne M par un sous-groupe de son groupe de similitudes Sim(M ). La démonstration des résultats de cette partie passe par l'étude des feuilletages munis d'une structure de similitude transverse. Sur ces feuilletages, on montre un résultat de rigidité qui peut être vu indépendamment des autres : ils sont soit transversalement plats, soit transversalement riemanniens. Remarquons que ces résultats sont valables dans le cas C ∞ , alors que de tels problèmes n'avaient été étudiés précédemment que dans le cas analytique.

Introduction (English)

This thesis is divided into three parts which may be read independently. In the first one, we study universality theorems for linkages whose ambiant space is a homogeneous surface. In the second one, we study the link between geodesic flows and billiards, as well as the dynamics of some linkages. The third one is about transverse similarity structures on foliations, and De Rham's decomposition theorem. Each of these parts contains its own introduction.

A linkage is a set of rigid rods joined together by hinges. Mathematically, one considers a linkage as a graph with an additional structure: lengths are given to the edges (the graph is said to be metric), and some vertices are fixed to the plane while the others move freely. A realization of a linkage in the plane is the choice of a position in the plane for each vertex, so that the edge lengths match with the distance in the plane between the corresponding vertices. In particular, one allows the edges to cross. Finally, the configuration space of a linkage is the set of all its realizations.

The first chapter is an introduction to the notion of linkage: we will present the historical background, and fundamental examples, which are useful for the next chapters.

The first part, after this introduction, is composed of four chapters and corresponds to the preprint [START_REF] Kourganoff | Universality theorems for linkages in homogeneous surfaces[END_REF]: we study linkages whose ambiant space is no longer the plane, but various Riemannian manifolds. Chapter 2 introduces the question of the universality of linkages: this notion corresponds to the idea that every curve would be traced out by a vertex of some linkage, and that any differentiable manifold would be the configuration space of some linkage. We shall present some results in this direction which are already known for planar linkages: on the one hand, the curves which may be traced out are exactly compact semi-algebraic curves; on the other hand, for any compact connected manifold M , there exists2 a configuration space of a linkage whose connected components are all diffeomorphic to M . The same chapter also contains the statements of all the main new results of this part, which are extensions of universality theorems to the Minkowski plane, the hyperbolic plane, and finally the sphere. In each case, one encounters different difficulties, and makes use of different techniques, but the results which are obtained are very similar, except in the Minkowski case, where the compacity hypothesis is no longer necessary: universality then becomes valid in a broader sense. The last three chapters contain the proofs of these statements, while the general tools are given in Chapter 2.

The second part is composed of three chapters, where the last two correspond to the preprint [START_REF] Kourganoff | Anosov geodesic flows, billiards and linkages[END_REF]. In Chapter 6, we establish a first link between geodesic flows on negatively curved manifolds and dispersive billiards, by putting in parallel the behaviors of these two systems. It is well-known since Sinai's work in the 1960's that these two CONTENTS systems are similar, but they are rarely studied together in the literature. In the same vein, we give a sufficient condition for a geodesic flow on a closed surface to be Anosov: it suffices that all solutions of the Ricatti equation along the geodesics, which equal zero at time t = 0, are greater than a single constant m > 0 at time t = 1. This well-known theorem has been used several times in the literature, but has apparently never been awarded any written proof: we will use it ourselves in the new results of this part. In Chapter 7, we present two new results concerning the geodesic flow of surfaces in the Euclidean R 3 , which undergo a strong contraction in one direction. Any surface in R 3 can be flattened with respect to the z-axis, and the flattened surface gets close to a billiard table in R 2 . We show that, under some hypotheses, the geodesic flow of the surface converges locally uniformly to the billiard flow. Moreover, if the billiard is dispersing, the chaotic properties of the billiard also apply to the geodesic flow: we show that it is Anosov in this case. Finally, in Chapter 8, we give generalities on the dynamics of linkages, and then apply the result of Chapter 7 to the theory of linkages. This provides a new example of Anosov linkage, made of 5 rods. It is the first time that one exhibits an Anosov linkage whose edge lengths are given explicitly. A video of this linkage, by Jos Leys, is available on my website.

The third part does not have a direct link with the two others, except for the study of Riemannian manifolds: it corresponds to the preprint [START_REF] Kourganoff | Transverse similarity structures on foliations, and de rham decomposition[END_REF]. We first consider manifolds with locally metric connections, that is, connections which are locally Levi-Civita connections of Riemannian metrics; we give in this framework an analog of De Rham's decomposition theorem, which usually applies to Riemannian manifolds. In the case such a connection also preserves a conformal structure, we show that this decomposition has at most two factors; moreover, when there are exactly two factors, one of them is the Euclidean space R q . Thus, we answer a question asked in [START_REF] Vladimir | Locally conformally berwald manifolds and compact quotients of reducible manifolds by homotheties[END_REF]. The study of locally metric connections which preserve a conformal structure is closely linked to "similarity structures" on manifolds: these are the structures obtained by the quotient of a Riemannian manifold M by a subgroup of its similarity group Sim(M ). The proofs of the results of this part use foliations with transverse similarity structures. On these foliations, we give a rigidity theorem of independant interest: they are either transversally flat, or transversally Riemannian. Notice that these results are valid in the C ∞ case, while such problems had only been studied in the analytic case previously.

Chapter 1

What is a linkage?

A mechanical linkage, or simply linkage, is a graph whose vertices are considered as rigid rods. Let us state precise mathematical definitions. Definition 1.1. A planar linkage L is a graph (V, E) together with:

1. A function l : E → R ≥0 (which gives the length of each edge); 2. A subset F ⊆ V of fixed vertices (represented by on the figures); 3. A function φ 0 : F → R 2 which indicates where the vertices of F are fixed. Definition 1.2. Let L be a planar linkage, and consider the Euclidean distance δ in R 2 . A realization of a planar linkage L is a function φ : V → R 2 such that:

1. For each edge (v 1 v 2 ) ∈ E, δ(φ(v 1 ), φ(v 2 )) = l(v 1 v 2 ); 2. φ| F = φ 0 .

The configuration space Conf(L) is the set of all realizations φ of L: it is naturally a subset of (R 2 ) n , where n is the number of vertices. It inherits the topology of the ambiant Euclidean space. Finally, the workspace of a vertex v ∈ V is the set {φ(v) | φ realization of L}. Linkages are one of the simplest physical examples involving manifolds of dimension 3 or more (other than the ambiant space R 3 ): they appear naturally as configuration spaces. In Section 1.1, we will give simple examples of linkages whose configuration space is diffeomorphic to T n or S n , for any natural number n. Generically, the dimension of the configuration space of a linkage is 2(|V | -|F |) -|E| (twice the number of free vertices, minus the number of edges). More precisely: Proposition 1.3. Choose any graph (V, E), any F ⊆ V and any φ 0 : F → R 2 . Then there is a set L of full Lebesgue measure in R E such that for all choice of edge lengths l ∈ L, the configuration space of L = (V, E, l, F, φ 0 ) is a smooth orientable manifold of dimension 2(|V | -|F |) -|E|.

Proof. Consider the function

F : (R 2 ) V \F → R E φ � → f φ
where, for any (vw) ∈ E and any φ ∈ (R 2 ) V \F , f φ (vw) = δ(φ(v), φ(w))

(here, the domain of φ is extended to the whole V using φ 0 ). Then for any l ∈ R E , the configuration space of L = (V, E, l, F, φ 0 ) is F -1 (l). By Sard's Theorem, the regular values of F form a set of full Lebesgue measure in R E . For such a value l, F -1 (l) is a smooth manifold of dimension 2(|V | -|F |) -|E|. Moreover, since R E is orientable, the normal bundle of F -1 (l) in (R 2 ) V \F is orientable as well. But (R 2 ) V \F itself is also orientable: one obtains an orientation of the tangent bundle of M , thus M is orientable.

Remark. Most of the linkages considered in Part I will not satisfy the assumptions of Proposition 1.3. In this case, the configuration space may still be a smooth manifold, but it does not need to be orientable, and it is impossible to compute its dimension from the number of edges and vertices alone.

There exist many natural problems involving linkages, which cover various fields of mathematics, such as algebraic geometry, algebraic topology, Riemannian geometry, dynamical systems, and the theory of computational complexity. Many interesting problems involving complexity may be found in [START_REF] Demaine | Geometric folding algorithms[END_REF]. In this thesis, we focus on two particular aspects of linkages: universality and dynamics. The examples in the rest of this chapter are chosen in view of these two problems.

Fundamental examples 1.The robotic arm

The robotic arm R n (Figure 1.2) is a linkage whose underlying graph is a path (all vertices have degree 2, except the two ends), with one fixed end. It has n edges of lengths l 1 , l 2 , . . . , l n .

The configuration space of R n is the torus T n : each of its configurations corresponds to n angles θ 1 , . . . , θ n formed by each of the n edges with the horizontal axis.

v 0 v 1 v 2 v 3 l 1 l 2 l 3 θ 1 θ 2 θ 3

Polygons

A polygon is a linkage without fixed vertices, whose underlying graph is a cycle.

It is often convenient to consider a polygon with two fixed adjacent vertices. In fact, fixing those two vertices amounts to removing the factor SO(2) � R 2 which is found in the configuration space of any linkage without fixed vertices.

After fixing these two vertices at a distance which corresponds to the edge between them, one may remove this edge which has become useless, without changing the configuration space. Thus, a polygon may be seen as a robotic arm whose two ends are fixed.

Example.

a b 1 1 1 1 1 1 1 1
Consider the linkage L n above with n edges. The vertices a and b are fixed at a distance n -� with a small enough � > 0, and all the edges have length 1.

Proposition 1.4. The configuration space Conf(L n ) is diffeomorphic to S n-2 .

Proof. We may assume that φ 0 (a) = (0, 0) and φ 0 (b) = (n -�, 0).

First, consider the robotic arm R n . For any configuration (θ 1 , . . . , θ n ) ∈ Conf(R n ), write s = (s 1 , s 2 ) = ( � cos θ i , � sin θ i ) ∈ R 2 the position of the last vertex v n . Then Conf(L n ) = {(θ 1 , . . . , θ n ) ∈ T n | s = (n -�, 0)}: for a small � > 0, Conf(L n ) is contained in an arbitrarily small neighborhood of C 0 = (0, . . . , 0). Now, consider the subset of all configurations whose last vertex lies on the horizontal axis:

E = {C ∈ T n | s 2 = 0}. Since the gradient of s 2 is ∇s 2 =    cos θ 1 . . . cos θ n   , E is a submanifold of T n in a neighborhood of C 0 .
Compute the differentials of s 1 up to order 1 and 2: Thus, Ds 1 (C 0 ) = 0 and D 2 s 1 (C 0 ) is non-degenerate. Moreover, s 1 reaches its global maximum n at C 0 . With Morse's Lemma, E is equipped with a global coordinate system (x i ) 1≤i≤n-1 near C 0 such that:

∇s 1 = -    sin θ 1 . . .
s 1 -n = - � n-1 � i=1 x 2 i � .
In particular, the level of E defined by s 1 = n -� is diffeomorphic to a sphere of dimension n -2 for a small enough � > 0.

This proof gives us a glimpse of how Morse theory may be used in view of determining the topology of some configuration spaces. The first chapter of [START_REF] Farber | Invitation to topological robotics[END_REF] uses this approach to describe the homology groups of the configuration spaces of polygons.

Spider linkages

Although less famous than polygons, spider linkages (with n legs) provide interesting examples and were studied by many authors.

A spider is made of a central vertex to which n legs are attached, each of which has one articulation (each leg is a copy of R 2 ). The end of each leg is fixed somewhere in the plane. For n = 2, the spider is in fact a pentagonal linkage.

Since a spider has 2n edges and n + 1 free vertices, its configuration space is a surface for a generic choice of the lengths of the edges: one may obtain a wide variety of surfaces in this way.

Spiders with n = 3. Thurston and Weeks [START_REF] Willam | The mathematics of three-dimensional manifolds[END_REF] detailed a particular case of a spider with n = 3, which they called triple linkage (Figure 1.3). In this case, each of the 3 legs of the spider restricts the movement of the central vertex to an annulus centered at a i , with inner radius |l 1l 2 | and outer radius l 1 + l 2 .

Thus, the workspace of the central vertex is the intersection of three annuli. When the lengths l 1 and l 2 vary, this intersection may take different shapes (Figure 1.4). Like Thurston and Weeks in their article, let us focus on the case1 on the left of Figure 1.4: the workspace of the central vertex x is a hexagon. For each position of x in the interior of the hexagon, there are two possible positions for p 1 , which are symmetric with respect to the line through x and a 1 . There are also two possible positions for each of the two other vertices, so any point in the interior of the workspace of the central vertex corresponds to 8 points in the configuration space. The boundary of the hexagon corresponds to configurations in which at least one of the arms is completely stretched or folded. Such configurations belong in fact to several hexagons. Thus, the configuration space of the linkage is made of 8 copies of the hexagon, glued together along their boundaries. Each edge belongs to two hexagons, and each vertex to four hexagons, so the polyhedron has 8 faces, 24 edges and 12 vertices. Its Euler characteristic is 8 -24 + 12 = -4, so2 it is diffeomorphic to a surface of genus 3.

For different choices of the edge lengths and positions of the fixed points, the intersection of the three annuli may take many forms: in each case, it is possible to make a similar computation to determine the genus of the surface (for example, it is possible to obtain the disjoint union of 6 spheres, or a surface of genus 12). Ten different topologies for the configuration space of the triple linkage are given in [START_REF] Hunt | Anosov parameter values for the triple linkage and a physical system with a uniformly chaotic attractor[END_REF].

The general case. In general, we have the following [START_REF] Mounoud | Sur l'espace des configurations d'une araignée[END_REF]: Theorem 1.5 (Mounoud, 2009). Let g be an natural number and r the biggest integer such that 2 r divides g -1. A compact orientable surface of genus g is diffeomorphic to a spider's configuration space if and only if (1/2 r )(g -1) ≤ 6r + 12.

In particular, it is impossible to realize a surface of genus 14 as a configuration space of a spider.

In 2006, O'Hara [O'H07] computed all the configuration spaces obtained by a spider whose arms all have the same length length (l 1 = l 2 = 1, with the notations of Figure 1.3), and whose fixed vertices are on the unit circle and form a regular polygon P in R 2 .

Theorem 1.6 (O' Hara, 2006). Let R be the radius of the circumscribed circle to the polygon P. There exists a critical value R n such that the configuration space is diffeomorphic to a connected orientable closed surface Σ g if R satisfies:

0 < R < 2 and R � = R n .
The genus g is given by

g = � 1 -2 n-1 + n2 n-3 + n2 n-1 = 1 + (5n -4)2 n-3 si 0 < R < R n , 1 -2 n-1 + n2 n-3 = 1 + (n -4)2 n-3 si R n < R < 2.
In his proof, O'Hara gives two methods to compute the genus: one of them is purely topological, while the other one uses Morse Theory. In the same paper, he also describes the singularities which appear when R does not satisfy the conditions of Theorem 1.6.

Centipedes

A centipede3 with n legs (Figure 1.5) is a linkage whose underlying graph has 2n + 1 vertices, where n + 1 free vertices form a path, and n fixed vertices are attached to the 1 st , 2 nd , . . . , (n -1) th and (n + 1) th vertex of the path, as in the following figure:

. . . with any edge lengths, and any positions for the fixed vertices. As for spiders, the configuration spaces of centipedes are generically surfaces. It is remarkable that any connected closed oriented surface is the configuration space of some centipede, as shown in [START_REF] Jordan | Compact surfaces as configuration spaces of mechanical linkages[END_REF].

In general, given a connected closed manifold M , it is unknown4 whether there exists a linkage whose configuration space is diffeomorphic to M . This kind of problem is called "universality problem": it is at the heart of Part I.

The pantograph

The pantograph (literally, a device which "writes everything" in Greek), invented by the astronomer Christoph Scheiner in 1603, was used to reproduce drawings at different In practice, a pen was fixed to the vertex C and the vertex B was moved along the drawing which was to be copied. For this linkage (among others), we will be interested in the possible positions of the two vertices B and C, rather than the topology of the configuration space.

Concerning the pantograph, two remarks are in order:

1. Here, we allowed some hinges (D and E) to be at the middle of bars, while our definition of a linkage as a mathematical graph requires them to be at the end.

We could change the definition to include this situation, but in our setting, it is more convenient to consider AD and DG as two different edges of length l, as well as another edge of length 2l between A and G. The three vertices A, D, G form a flat triangle, so they are aligned for all configurations. With this technique, it is possible to add a hinge anywhere on a bar.

2. With our definitions, there are in fact many realizations of this linkage such that C is not the image of B by a homothety of center A. For example, for any position of A, C, G, there is a realization such that B = G. This is known as the problem of degenerated configurations: they have to be dealt with carefully when trying to understand the topology of configuration spaces. The same problem will occur in Section 1.2.2. For more details, see [START_REF] Kapovich | Universality theorems for configuration spaces of planar linkages[END_REF] or Chapter 3.

Straight-line motion

The problem of straight-line motion appeared naturally when Watt designed his doubleaction steam engine in 1781. He needed a mechanism able to guide the piston of the engine along a straight line, and to transmit the energy to other elements of the system (for example, a wheel). With our definitions, the question was the following:

Question. Does there exist a linkage containing one vertex whose workspace is a line segment?

1 2 3 4 5
Figure 1.7 -Newcomen's steam engine 5 . The cylinder (on the right) is filled with steam while the pump (on the left) is pulled down by its own weight [START_REF]) ( i.e. U = M and φ is a bijection), the manifold M is isometric to the Euclidean R q for some q ≥ 0. Proof. Let us prove the first statement. Choose any y ∈ M and four vectors a, b, c, d in T y M of unit length for g. The point is that φ preserves R[END_REF]. Then cold water is injected into the cylinder (3), which condensates the steam, creates vacuum and lowers the piston (4): at the other side, the pump goes up and takes the water out from the mine.

Earlier steam engines did not require such a mechanism. Half a century before Watt, Newcomen designed another steam engine, which was widely used to pump water from the coal mines. The steam only pulled the piston to one side (contrary to Watt's double-action engine, where steam pulled it alternatively to both sides), and the mass of the pump on the other side pulled the piston back to its original position (Figure 1.7). In Newcomen's engine, it was possible to achieve straight-line motion with a simple flexible chain. In contrast, Watt needed a rigid linkage to guide the piston.

Watt's linkage

Watt's linkage (Figure 1.8) contains one vertex whose workspace is close to a straight line. It was used in Watt's famous double-action steam engine, and is still used in the suspension systems of some cars.

For engineers, the problem of straight-line motion was solved, but for mathematicians, it was only the beginning.

Elementary computation shows that the workspace of the central vertex is the curve of equation (in polar coordinates):

r 2 = b 2 -(a sin θ ± � c 2 -a 2 cos 2 θ) 2 .
It is called Watt's curve and, with a good choice of parameters, it has the shape of an eight (like on the figure). Near the center, it has curvature 0, so it is a straight line up to order 2, which is sufficient for most practical applications.

The Peaucellier inversor

In the 1860's, Peaucellier and Lipkin discovered simultaneously a linkage which achieved perfect straight-line motion. First, we introduce the Peaucellier inversor (Figure 1.9).

, BH 2 = l 2 -DH 2 = r 2 -AH 2 . Thus, AH 2 -DH 2 = r 2 -l 2 , so AD • AE = r 2 -l 2 .
In Chapters 3 and 4, this linkage will be adapted to other ambiant geometries. The workspace of D is contained in a circle C centered at G, so the workspace of E is contained in the image of C by the inversion with respect to the circle centered at A, of radius r 2l 2 . If one chooses the position of G and the length of the new edge s so that A ∈ C, then the image of C by an inversion centered at A is a straight line. Therefore, the workspace of E is contained in a straight line (more precisely, it is a line segment).

A popular catchphrase is the following: "The Peaucellier linkage transforms linear motion into circular motion." Indeed this is true in some sense, since D's workspace is contained in a circle, while E's workspace is a line segment. However, this formulation might let think that, in a steam engine, D corresponds to a wheel and E to the piston, which cannot be the case: the vertex D does not follow a whole circle, but only goes back and forth on a circular arc! In fact, the only important fact in the Peaucellier linkage is that one vertex follows a straight line. Once this goal is achieved, is it possible to transmit the energy to a wheel using simply one bar, fixed somewhere on the wheel (Figure 1.11).

Hart's linkage

Other straight-line mechanisms

Many other mathematicians discovered linkages which provide approximate or exact linear motion, including Chebyshev, Kempe and Sylvester (see [START_REF] Alfred B Kempe | How to draw a straight line[END_REF] for a detailed review of such linkages).

Very recently, the Dutch artist Theo Jansen designed his own approximate straight-line motion linkage. It allows his large "creatures" made of plastic rods to "walk" smoothly on the beach. Part I

Universality theorems for linkages in homogeneous surfaces

Chapter 2

Introduction and generalities on universality

Throughout Part I, we shall consider linkages which are not necessarily planar: the ambiant space may be any manifold M instead of R 2 . A realization of a linkage L in a manifold M is a mapping which sends each vertex of the graph to a point of M, respecting the lengths of the edges. The configuration space Conf M (L) is the set of all realizations of L in M. This supposes, classically, the ambient manifold M to have a Riemannian structure: thus the configuration space may be seen as the space of "isometric immersions" of the metric graph L in M.

Here we will always deal with (non-trivially) marked connected graphs, that is, a nonempty set of vertices have fixed realizations (in fact, when M is homogeneous, considering a linkage without fixed vertices only adds a translation factor to the configuration space). Hence, our configurations spaces will be compact even if M is not compact, but rather complete.

Some historical background

Most existing studies deal with the special case where M is the Euclidean plane and some with the higher dimensional Euclidean case (see for instance [START_REF] Farber | Invitation to topological robotics[END_REF] and [START_REF] Henry | Configuration spaces of linkages in R n[END_REF]). There are also studies about polygonal linkages in the standard 2-sphere (see [KM + 99]), or in the hyperbolic plane (see [START_REF] Kapovich | The relative deformation theory of representations and flat connections and deformations of linkages in constant curvature spaces[END_REF]).

Universality theorems. When M is the Euclidean plane E 2 , a configuration space is an algebraic set. This set is smooth for a generic length structure on the underlying graph.

Universality theorems tend to state that, playing with mechanisms, we get any algebraic set of R n , and any manifold, as a configuration space! In contrast, it is a hard task to understand the topology or geometry of the configuration space of a given mechanism, even for a simple one.

Universality theorems have been announced in the ambient manifold E 2 by Thurston in oral lectures, and then proved by Kapovich and Millson in [KM02]. They have been proved in E n by King [START_REF] Henry | Configuration spaces of linkages in R n[END_REF], and in RP 2 and in the 2-sphere by Mnëv (see [START_REF] Mnëv | The universality theorems on the classification problem of configuration varieties and convex polytopes varieties[END_REF] and [START_REF] Kapovich | Universality theorems for configuration spaces of planar linkages[END_REF]). It is our aim in Part I to prove them in the cases of: the hyperbolic plane H 2 , the sphere S 2 and the (Lorentz-)Minkowski plane M. These are simply connected homogeneous pseudo-Riemannian surfaces (the list of such spaces includes in addition the Euclidean and the de Sitter planes). Then it becomes natural to ask whether universality theorems hold in a more general class of manifolds, for instance on Riemannian surfaces without a homogeneity hypothesis.

In order to be more precise, it will be useful to introduce partial configuration spaces: for W a subset of the vertices of L, one defines Conf W M (L) as the set of realizations of the subgraph induced by W that extend to realizations of L. One has in particular a restriction map: Conf M (L) → Conf W M (L). If W = {a} is a vertex of L, its partial configuration space is its workspace, i.e. the set of all its positions in M corresponding to realizations of L.

Euclidean planar linkages. Now regarding the algebraic side of universality, the history starts (and almost ends) in 1876 with the well-known Kempe's theorem [START_REF] Kempe | On a General Method of describing Plane Curves of the nth degree by Linkwork[END_REF]: Theorem 2.1. Any algebraic curve of the Euclidean plane E 2 , intersected with a Euclidean ball, is the workspace of some vertex of some mechanical linkage.

This theorem has the following natural generalization, which we will call the algebraic universality theorem, proved by Kapovich and Millson (see [START_REF] Kapovich | Universality theorems for configuration spaces of planar linkages[END_REF]):

Theorem 2.2. Let A be a compact semi-algebraic subset (see Definition 2.12) of (E 2 ) n (identified with R 2n ). Then, A is a partial configuration space Conf W E 2 (L) of some linkage L in E 2 . When A is algebraic, one can choose L such that the restriction map Conf E 2 (L) → A = Conf W E 2 (L) is a smooth finite trivial covering.
When Conf E 2 (L) is not a smooth manifold, as usual, by a smooth map on it, we mean the restriction of a smooth map defined on the ambient R 2n .

From Theorem 2.2, Kapovich and Millson easily derive the differential universality theorem on the Euclidean plane: Theorem 2.3. Any compact connected smooth manifold is diffeomorphic to one connected component of the configuration space of some linkage in the Euclidean plane E 2 . More precisely, there is a configuration space whose components are all diffeomorphic to the given differentiable manifold.

Jordan and Steiner also proved a weaker version of this theorem with more elementary techniques (see [START_REF] Jordan | Configuration spaces of mechanical linkages[END_REF]).

How to go from the algebraic universality to the differentiable one? The differentiable universality theorems (Theorems 2.3, 2.5 and 2.7) follow immediately from the algebraic ones (Theorems 2.2, 2.4 and 2.6) once we know which smooth manifolds are diffeomorphic to algebraic sets. In 1952, Nash [START_REF] Nash | Real algebraic manifolds[END_REF] proved that for any smooth connected compact manifold M , one may find an algebraic set which has one component diffeomorphic to M . In 1973, Tognoli [START_REF] Tognoli | Su una congettura di nash[END_REF] proved that there is in fact an algebraic set which is diffeomorphic to M (a proof may be found in [START_REF] Akbulut | Topology of real algebraic sets[END_REF], or in [START_REF] Bochnak | Real algebraic geometry[END_REF]).

In the non-compact case (in which we will be especially interested), Akbulut and King [START_REF] Akbulut | The topology of real algebraic sets with isolated singularities[END_REF] proved that every smooth manifold which is obtained as the interior of a compact manifold (with boundary) is diffeomorphic to an algebraic set. Note that conversely, any (non-singular) algebraic set is diffeomorphic to the interior of a compact manifold with boundary.

Results

It is very natural to ask if these algebraic and differential universality theorems can be formulated and proved for configuration spaces in a general target space M. Our results suggest this could be true: indeed, we naturally generalize universality theorems to the cases of M = M, H 2 and S 2 , the Minkowski and hyperbolic planes and the sphere, respectively. Notice that for a general M, there is no notion of algebraic subset of M n ! We will however observe that there is a natural one in the cases we are considering here. In the general case, the question around Kempe's theorem could be rather formulated as: "Characterize curves in M that are workspaces of some vertex of a linkage."

Minkowski planar linkages. These linkages are studied in Chapter 3. Classically, the structure of M needed to define realizations of a linkage is that of a Riemannian manifold. Observe however that a distance, not necessarily of Riemannian type, on M would also suffice for this task. But our idea here is instead to relax positiveness of the metric. Instead of a Riemannian metric, we will assume M has a pseudo-Riemannian one. We will actually restrict ourselves to the simple flat case where M is a linear space endowed with a non-degenerate quadratic form, and more specially to the 2-dimensional case, that is the Minkowski plane M. On the graph side, weights of edges are no longer assumed to be positive numbers. This framework extension is mathematically natural, and may be related to the problem of the embedding of causal sets in physics, but the most important (as well as exciting) fact for us is that configuration spaces are (a priori) no longer compact, and we want to see what new spaces we get in this new setting.

The Lorentz-Minkowski plane M is R 2 endowed with a non-degenerate indefinite quadratic form. We denote the "space coordinate" by x and the "time coordinate" by t.

The configuration space Conf M (L) is an algebraic subset (defined by polynomials of degree 2) of M n = R 2n (n is the number of vertices of L), and similarly a partial configuration space Conf W M (L) is semi-algebraic (see Definiton 2.12). In contrast to the Euclidean case, these sets may be non-compact (even if L has some fixed vertices in M). We will prove: Theorem 2.4. Let A be a semi-algebraic subset of M n (identified with R 2n ). Then, A is a partial configuration space Conf W M (L) of some linkage L in M. When A is algebraic, one can choose L such that the restriction map Conf M (L) → A is a smooth finite trivial covering.

Somehow, considering Minkowskian linkages is the exact way of realizing non-compact algebraic sets! In particular, Kempe's theorem extends (globally, i.e. without taking the intersection with balls) to the Minkowski plane: any algebraic curve is the workspace of one vertex of some linkage.

Remark. If the restriction map is injective, then it is a bijective algebraic morphism from Conf M (L) to A, but not necessarily an algebraic isomorphism. In fact, it is true for non-singular complex algebraic sets that bijective morphisms are isomorphisms, but this is no longer true in the real algebraic case (see for instance [START_REF] Mumford | Algebraic geometry. I. Classics in Mathematics[END_REF], Chapter 3).

We also have a differential version of the universality theorem in the Minkowski plane (which follows directly from Theorem 2.4, as explained at the end of Section 2.1): Theorem 2.5. For any differentiable manifold M with finite topology, i.e. diffeomorphic to the interior of a compact manifold with boundary, there is a linkage in the Minkowski plane with a configuration space whose components are all diffeomorphic to M . More precisely, there is a partial configuration space Conf W M (L) which is diffeomorphic to M and such that the restriction map Conf M (L) → Conf W M (L) is a smooth finite trivial covering.

Hyperbolic planar linkages. In Chapter 4, we prove that both algebraic and differential universality theorems hold in the hyperbolic plane. The problem is that the notion of algebraic set has no intrinsic definition in the hyperbolic plane. However, it is possible to define an algebraic set in the Poincaré half-plane model Conversely, any partial configuration space of any linkage with at least one fixed vertex is a compact semi-algebraic subset of (H 2 ) n , so this theorem characterizes the sets which are partial configuration spaces (see Definiton 2.12 for the notion of "semi-algebraic").

�� x y � ∈ R 2     y > 0 � (
In particular, Kempe's theorem holds in the hyperbolic plane.

And here follows the differential version:

Theorem 2.7. For any compact differentiable manifold M , there is a linkage in the hyperbolic plane with a configuration space whose components are all diffeomorphic to M . More precisely, there is a partial configuration space Conf W H 2 (L) which is diffeomorphic to M and such that the restriction map

Conf H 2 (L) → Conf W H 2 (L) is a smooth finite trivial covering.
Spherical linkages. These linkages are the subject of Chapter 5. In 1988, Mnëv [START_REF] Mnëv | The universality theorems on the classification problem of configuration varieties and convex polytopes varieties[END_REF] proved that the algebraic and differential universality theorems hold true in the real projective plane RP 2 endowed with its usual metric as a quotient of the standard 2-sphere. Even better, he showed that the number of copies in the differential universality for RP 2 can be reduced to 1, i.e. any manifold is the configuration space of some linkage. As Kapovich and Millson pointed out [START_REF] Kapovich | Universality theorems for configuration spaces of planar linkages[END_REF], a direct consequence of Mnëv's theorem is the differential universality theorem for the 2-sphere (but, this time, we get several copies of the desired manifold): Theorem 2.8 (Mnëv-Kapovich-Millson). For any compact differentiable manifold M , there is a linkage in the sphere with a configuration space whose components are all diffeomorphic to M .

However, it seems impossible to use Mnëv's techniques to prove the algebraic universality for spherical linkages: for example, all the configuration spaces of his linkages are symmetric with respect to the origin of R 3 . In order to obtain any semi-algebraic set as a partial configuration space, we need to start again from scratch and construct linkages specifically for the sphere.

Contrary to the Minkowski and hyperbolic cases, the generalization of the theorems to higher dimensional spheres is straightforward. Thus, we are able to prove the following: Theorem 2.9. Let d ≥ 2 and let A be a compact semi-algebraic subset of (S d ) n (identified with a subset of R (d+1)n ). Then, A is a partial configuration space of some linkage L in S d .

In particular, Kempe's theorem holds in the sphere. Conversely, any partial configuration space of any linkage is a compact semi-algebraic subset of (S d ) n (see Section 2.4), so this theorem characterizes the sets which are partial configuration spaces.

Let us note that even when A is algebraic, our construction does not provide a linkage L such that the restriction map Conf S d (L) → A is a smooth finite trivial covering. We do not know whether such a linkage exists. Some questions. Our results suggest naturally -among many questions -the following:

1. Besides the 2-dimensional case, are the results in the Minkowski plane true for any (finite-dimensional) linear space endowed with a non-degenerate quadratic form? And what about higher-dimensional hyperbolic spaces? It is likely that the adaptation of the 2-dimensional proof hides no surprise, like in the Euclidean case, but it would probably require tedious work to prove it.

2. In our definition of linkages in the Minkowski plane, we allow some edges to have imaginary lengths (they are "timelike"). Is it possible to require the graphs of Theorems 2.4 and 2.5 to be spacelike, i.e. require all their edges to have real lengths?

3. In all the universality theorems that we prove, we obtain a linkage whose configuration space is diffeomorphic to the sum of a finite number of copies of the given manifold M . Is it possible to choose this sum trivial, that is, with exactly one copy of M ? (This question is also open in the Euclidean plane.)

4. Is the differential universality theorem true on any Riemannian manifold?

Linkages on Riemannian manifolds. Let us give a partial answer to the last question using the following idea: just as the surface of the earth looks flat to us, any Riemannian manifold will almost behave as the Euclidean space if one considers a linkage which is small enough. However, our linkage has to be robust to small perturbations of the lengths, which is not the case for many of the linkages described in Part I (consider for example the rigidified square linkage).

Theorem 2.10. Consider a linkage L in the Euclidean space E n , with at least one fixed vertex, such that for any small perturbation of the length vector l ∈ (R ≥0 ) E , the configuration space Conf E n (L) remains the same up to diffeomorphism. Then for any Riemannian manifold M, there exists a linkage L M in M whose configuration space is diffeomorphic to Conf E n (L).

In particular, Theorem 2.10 combined with the work of Jordan and Steiner [JS01] yields directly Corollary 2.11. In any Riemannian surface M, the differentiable universality theorem is true for compact orientable surfaces. In other words, any compact orientable surface is diffeomorphic to the configuration space Conf M (L) of some linkage L.

This leads to the following Question. Which manifolds can be obtained as the configuration space of some linkage in R n which is robust to small perturbations (in the sense of Theorem 2.10) ?

This question is probably very difficult, but it is clear that there are restrictions on such manifolds: for example, they have to be orientable (because of Proposition 1.3).

Ingredients of the proofs

There are essentially three technical as well as conceptual tools: functional linkages, combination of elementary linkages, and regular inputs. The main idea is always the same as in all the known proofs of Universality theorems (see the proofs of Thurston, Mnëv [START_REF] Mnëv | The universality theorems on the classification problem of configuration varieties and convex polytopes varieties[END_REF], King [START_REF] Henry | Configuration spaces of linkages in R n[END_REF] or Kapovich and Millson [KM02]): one combines elementary linkages to construct a "polynomial linkage". Functional linkages. One major ingredient in the proofs is the notion of functional linkages. Here we enrich the graph structure by marking two new vertex subsets P and Q playing the role of inputs and outputs, respectively. If the partial realization of Q is determined by the partial realization of P , by means of a function f : Conf P M (L) → M Q (called the input-output function), then we say that we have a functional linkage for f (for us, M will be the Minkowski plane M, the hyperbolic plane H 2 or the sphere S d ). The Peaucellier linkage is a famous historical example: it is functional for an inversion with respect to a circle. With the notations of Figure 1.9, the input is D and the output is E.

Combination. Another major step in the proofs consists in proving the existence of functional linkages associated to any given polynomial f . This will be done by "combining" elementary functional linkages. We define combination so that combining two functional linkages for the functions f 1 and f 2 provides a functional linkage for

f 1 • f 2 .
Elementary linkages. All the work then concentrates in proving the existence of linkages for suitable elementary functions (observe that even for elementary linkages one uses a combination of more elementary ones). As an example, we give the list of the elementary linkages needed to prove Theorem 2.4 (in the Minkowski case):

1. The linkages for geometric operations:

(a) The robotic arm linkage (Section 3.2.1): one of the most basic linkages, used everywhere in our proofs and in robotics in general.

(b) The rigidified square (Section 3.2.2): a way of getting rid of degenerate configurations of the square using a well-known construction.

(c) The Peaucellier inversor (Section 3.2.3): this famous linkage of the 1860's has a slightly different behavior in the Minkowski plane but achieves basically the same goal.

(d) The partial t 0 -line linkage (Section 3.2.4): it is obtained using a Peaucellier linkage, but does not trace out the whole line.

(e) The t 0 -integer linkage (Section 3.2.5): it is a linkage with a discrete configuration space.

(f) The t 0 -line linkage (Section 3.2.6): it draws the whole line, and is obtained by combining the two previous linkages.

(g) The horizontal parallelizer (Section 3.2.7): it forces two vertices to have the same ordinate, and it is obtained by combining several line linkages.

(h) The diagonal parallelizer (Section 3.2.8): its role is similar to the horizontal parallelizer but its construction is totally different.

2. The linkages for algebraic operations, which realize computations on the t = 0 line:

(a) The average function linkage (Section 3.3.1): it computes the average of two numbers, and is obtained by combining several of the previous linkages.

(b) The adder (Section 3.3.2): it is functional for addition on the t = 0 line, and is obtained from several average function linkages.

(c) The square function linkage (Section 3.3.3): it is functional for the square function and is obtained by combining the Peaucellier linkage (which is functional for inversion) with adders. This linkage is somewhat difficult to obtain because we want the inputs to be able to move everywhere in the line, while the inversion is of course not defined at x = 0.

(d) The multiplier (Section 3.3.4): it is functional for multiplication and is obtained from square function linkages.

(e) The polynomial linkage (Section 3.3.5): obtained by combining adders and multipliers, it is functional for a given polynomial function f . This linkage is used to prove the universality theorems: if the outputs are fixed to 0, the inputs are allowed to move exactly in f -1 (0).

Regular inputs.

In our theorems, we need the restriction map Conf M (L) → Conf P M (L) to be a smooth finite trivial covering. In the differential universality Theorem, it ensures in particular that the whole configuration space consists in several copies of the given manifold M . The set of regular inputs Reg P M (L) is the set of all realizations of the inputs which admit a neighborhood onto which the restriction map is a smooth finite covering. We have to be very careful, because even for quite simple linkages such as the robotic arm, the restriction map is not a smooth covering everywhere! There are mainly two possible reasons for the restriction map not to be a smooth covering:

1. One realization of the inputs may correspond to infinitely many realizations of the whole linkage (for example, when the robotic arm in Section 3.2.1 has two inputs fixed at the same location, the workspace of the third vertex is a whole circle).

2. Even if it corresponds only to a finite number of realizations, these realizations may not depend smoothly on the inputs (for example, when the robotic arm in Section 3.2.1 is stretched).

New difficulties in each case. While the idea is always the same in all known proofs of universality theorems for linkages, i.e. combine elementary linkages to form a functional linkage for polynomials, each case has its own new difficulties due to different geometric properties, and the elementary linkages always require major changes to work correctly.

Here follow examples of such differences with the Euclidean case:

The Minkowski case

1.
The Minkowski plane M is not isotropic: its directions are not all equivalent. Indeed, these directions have a causal character in the sense that they may be spacelike, lightlike or timelike. For example, one needs different linkages in order to draw spacelike, timelike and lightlike lines.

2. In the Euclidean plane, two circles C(x, r) and C(x � , r � ) intersect if and only if |rr � | ≤ �xx � � ≤ r + r � , but in the Minkowski plane, the condition of intersection is much more complicated to state (see Section 3.1.2).

3. In the Euclidean plane, one only has to consider compact algebraic sets. Applying a homothety, one may assume such a set to be inside a small neighborhood of zero, which makes the proof easier. Here, the algebraic sets are no longer compact, so we have to work with mechanisms which are able to deal with the whole plane.

The hyperbolic case 1. The rigidified square linkage, used extensively in all known proofs in the flat case, does not work anymore in its usual form, and does not have a simple analogue.

2. There is no natural notion of homothety: in particular, the pantograph does not compute the middle of a hyperbolic segment, contrary to the flat cases.

3. The notion of algebraic set is less natural than in the flat case.

4. In every standard model of the hyperbolic plane (such as the Poincaré half-plane), the expression of the distance between two points is much more complicated than in the flat case.

The spherical case 1. Just as in the hyperbolic case, the curvature prevents the rigidified square linkage from working correctly.

2. There is no natural notion of homothety.

3. In the Euclidean or hyperbolic planes, we only need to prove algebraic universality for bounded algebraic sets, which means that our functional linkages do not need to work on the whole surface. In the sphere, all the distances are uniformly bounded (even the lengths of the edges of our linkages), so we need to take into account the whole sphere when constructing linkages.

4. The compactness of the sphere also makes it difficult to construct linkages which deal with algebraic operations (addition, multiplication, division) since there is no proper embedding of R in the sphere.

Algebraic and semi-algebraic sets

In this section, we recall the standard definitions of algebraic and semi-algebraic sets.

We adapt them to the Minkowski plane, the hyperbolic plane and the sphere in a natural way and state some of their properties.

Definition 2.12. An algebraic subset of R n is a set A ⊆ R n such that there exist m ∈ N and f : R n → R m a polynomial such that A = f -1 (0). We define a semi-algebraic subset of R n (see [START_REF] Bochnak | Real algebraic geometry[END_REF]) as the projection of an algebraic set 1 . More precisely, it is a set B such that there exists N ≥ n and an algebraic set A of R N such that B = π(A), where π is the projection onto the first coordinates

π : R N = R n × R N -n → R n (x, y) � -→ x.
We define the (semi-)algebraic subsets of M n by identifying M n with (R 2 ) n = R 2n . We also define the (semi-)algebraic subsets of (H 2 ) n , using the Poincaré half-plane model (see Definition 4.1), as the (semi-)algebraic subsets of R 2n which are contained in ��

x y

� ∈ R 2     y > 0 � n .
Finally, a (semi-)algebraic subset of (S d ) n (for d ≥ 2) is a semi-algebraic subset of R d+1 which is contained in the unit sphere of R d+1 . Proposition 2.13. For any compact semi-algebraic subset B of R n , there exists N ≥ n and a compact algebraic subset A of R N such that B = π(A), where π is the projection onto the first coordinates: R N → R n .

Proof. First case. Assume for the moment that there exist polynomials f 1 , . . . , f m : R n → R such that

B = {x ∈ R n | ∀i ∈ {1, . . . , m} f i (x) ≥ 0} . Let h : R n+m = R n × R m → R m   x,    y 1 . . . y m       � -→    f 1 (x) -y 2 1 . . . f m (x) -y 2 m   
and A = h -1 (0). Then the projection of A onto the first n coordinates is obviously B. Moreover, A is compact since it is the image of B by the continuous function

g : B → R n+m = R n × R m x � -→   x,    � f 1 (x) . . . � f m (x)       .
General case. The finiteness theorem for semi-algebraic sets (see [START_REF] Bochnak | Real algebraic geometry[END_REF], 2.7.2) states that any closed algebraic set can be described as the union of a finite number of sets B 1 , . . . , B k which satisfy the assumption of the first case: apply the first case to each of the B i 's to end the proof.

We end this section with two analogous propositions for the hyperbolic plane and the sphere.

Proposition 2.14. For any compact semi-algebraic subset B of (H 2 ) n , there exists N ≥ n and a compact algebraic subset A of (H 2 ) N (with some N ≥ n) such that B = π(A), where π is the projection onto the first coordinates:

(H 2 ) N → (H 2 ) n . Proof. Let A � be a compact algebraic set of R N � (with some N � ≥ 2n) such that B = π(A � ),
where π is the projection onto the first coordinates: R N � → R 2n . Then the projection of the compact algebraic set

A := ��� x1 y1 � , . . . , � xn yn � , � xn+1 1 � , . . . , � x N � -n 1 ��     (x1, y1, . . . , xn, yn, xn+1, . . . , x N � -n ) ∈ A � � (where A ⊆ (H 2 ) N � -n ) is exactly B.
Proposition 2.15. For any compact semi-algebraic subset B of (S 2 ) n , there exists N ≥ n and a (compact) algebraic subset A of (S 2 ) N (with some N ≥ n) such that B = π(A), where π is the projection onto the first coordinates:

(S 2 ) N → (S 2 ) n . Proof. Let A � be a compact algebraic set of R N � (with some N � ≥ 3n) such that B = π(A � ),
where π is the projection onto the first coordinates:

R N � → R 3n . Since A � is compact, there is a λ such that π 2 (A � ) ∈ [-λ, λ] N � -3n
, where π 2 is the projection onto the last coordinates: R N � → R N � -3n . Then the projection of the compact algebraic set

A :=          x 1 y 1 z 1   , . . . ,   xn yn zn   ,   x n+1 y n+1 0   , . . . ,   x N � -2n y N � -2n 0            � x 1 , y 1 , z 1 , . . . , xn, yn, zn, λx n+1 , . . . , λx N � -2n � ∈ A � x 2 n+1 + y 2 n+1 = 1, . . . , x 2 N � -2n + y 2 N � -2n = 1      (where A ⊆ (S 2 ) N � -2n ) is exactly B.
Of course, Proposition 2.15 extends to S d with any d ≥ 2.

Generalities on linkages

In the present section, we develop generalities on linkages which apply to the Minkowski plane, the hyperbolic plane and the sphere. Thus, we consider a smooth manifold M endowed with a distance function

δ : M × M → R ≥0 ∪ iR ≥0 .
In the case of a Riemannian manifold (in particular, for the hyperbolic plane and the sphere), the metric determines a real-valued distance on M.

In the case of the Minkowski plane, M is the plane R2 . Here, we argue by a naive algebraic analogy and define a distance as

δ �� x 1 t 1 � , � x 2 t 2 �� = � (x 2 -x 1 ) 2 -(t 2 -t 1 ) 2 ∈ R ≥0 ∪ iR ≥0 .
Accordingly, the length structure of the linkage will be generalized by taking values in R ≥0 ∪ iR ≥0 (instead of R ≥0 ) as follows: When the linkage is named L 1 , we usually write

Definition 2.16. A linkage L in M is a graph (V, E) together with: 1. A function l : E → R ≥0 ∪ iR ≥0 (
L 1 = (V 1 , E 1 , l 1 , . . . ) and name its vertices a 1 , b 1 , c 1 , . . . . If the linkage L 1 is a copy of the linkage L, the vertex a 1 ∈ V 1
corresponds to the vertex a ∈ V , and so on.

Definition 2.17. Let L be a linkage in M. A realization of a linkage L in M is a function φ : V → M such that: 1. For each edge v 1 v 2 ∈ E, δ(φ(v 1 ), φ(v 2 )) = l(v 1 v 2 ); 2. φ| F = φ 0 .
Remark. On the figures of Part I, linkages are represented by abstract graphs. The edges are not necessarily represented by straight segments, and the positions of the vertices on the figures do not necessarily correspond to a realization (unless otherwise stated).

Definition 2.18. Let L be a linkage in M. Let W ⊆ V . The partial configuration space of L in M with respect to W is Conf W M (L) = {φ| W | φ realization of L} .
In other words, Conf W M (L) is the set of all the maps φ : W → M which extend to realizations of L. In particular, the configuration space

Conf M (L) = Conf V M (L) is the set of all realizations of L.
Definition 2.19. A marked linkage is a tuple (L, P, Q), where P and Q are subsets of V : P is called the "input set" and its elements, called the "inputs", are represented by on the figures, whereas Q is called the "output set" and its elements, called the "outputs", are represented by on the figures.

The input map p : Conf M (L) → M P is the map induced by the projection M V → M P (the restriction map). In other words, for all φ ∈ Conf M (L), we have p(φ) = φ| P .

Likewise, we define the output map q

: Conf M (L) → M Q by q(φ) = φ| Q .
The notion of marked linkage is not necessary to study configuration spaces. However, in the linkages we use in our proofs, some vertices play an important role (the inputs and the outputs) while others do not: this is why we always consider marked linkages. The following notion3 accounts for the names "inputs" and "outputs": Definition 2.20. We say that L is a functional linkage for the input-output function

f : Conf P M (L) → M Q if ∀φ ∈ Conf M (L) f (p(φ)) = q(φ).

Regularity

Definition 2.21. Let L be a linkage. Let W ⊆ V and ψ ∈ Conf P M (L). Let π W be the restriction map

π W : Conf W ∪P M (L) → Conf P M (L). We say that ψ is a regular input for W if there exists an open neighborhood U ⊆ Conf P M (L) of ψ such that π W | π -1 W (U ) is a finite smooth covering 4 . We write Reg P M (L, W ) ⊆ Conf P M (L)
the set of regular inputs for W . When W is the set V of all vertices, we simply write Reg P M (L). Roughly speaking, ψ is a regular input for W if it determines a finite number of realizations φ of W , and if these configurations are determined smoothly with respect to ψ (in other words, π -1 W is a smooth multivalued function in a neighborhood of ψ). The following fact is simple but essential: Fact 2.22. For any W 1 , W 2 ⊆ V , we have

Reg P M (L, W 1 ) ∩ Reg P M (L, W 2 ) ⊆ Reg P M (L, W 1 ∪ W 2 )
. Therefore, in practice, in order to prove that Reg P M (L) = Conf P M (L), we only have to prove that Reg P M (L, {v}) = Conf P M (L) for all v ∈ V .

Changing the input set

In this proposition, we take a linkage, then consider the same linkage with a different set of inputs P and analyse the impact on Reg

P M (L). Fact 2.23. Let L 1 = (V 1 , E 1 , l 1 , F 1 , φ 01 , P 1 , Q 1 ), P 2 ⊆ V 1 and define L 2 = (V 1 , E 1 , l 1 , F 1 , φ 01 , P 2 , Q 1 ). Recall that p 1 : Conf M (L 1 ) → M P 1 and p 2 : Conf M (L 2 ) → M P 2 are the respective input maps of L 1 and L 2 . Then Reg P 2 M (L 2 ) contains � ψ ∈ Conf P 2 M (L 2 )    ∀φ ∈ p -1 2 (ψ) p 1 (φ) ∈ Reg P 1 M (L 1 ) � ∩ Reg P 2 M (L 2 , P 1 ).
Proof. This is a simple consequence of the fact that the composition of two smooth functions is a smooth function.

Combining linkages

This notion is essential to construct complex linkages from elementary ones. The proofs in this section are straightforward and left to the reader.

Let

L 1 = (V 1 , E 1 , l 1 , F 1 , φ 01 , P 1 , Q 1 ) and L 2 = (V 2 , E 2 , l 2 , F 2 , φ 02 , P 2 , Q 2 ) be two linkages, W 1 ⊆ V 1 , and β : W 1 → V 2 .
The idea is to construct a new linkage L 3 = L 1 ∪ β L 2 as follows:

Step

1 Consider L 1 ∪ L 2 , the disjoint union of the two graphs (V 1 , E 1 ) and (V 2 , E 2 ).
Step 2 Identify some vertices of V 1 with some vertices of V 2 via β, without removing any edge.

Since linkages are graphs which come with an additional structure, we need to clarify what happens to the other elements (l, F , φ 0 , P , Q). In particular, note that the inputs of L 2 which are in β(W 1 ) are not considered as inputs in the new linkage L 3 . Definition 2.24 (Combining two linkages). We define

L 3 = L 1 ∪ β L 2 = (V 3 , E 3 , l 3 , F 3 , φ 03 , P 3 , Q 3 )
in the following way:

1. V 3 = (V 1 \ W 1 ) ∪ V 2 ; 2. E 3 = (E 1 ∩ (V 1 \ W 1 ) 2 ) ∪ (E 2 ∩ V 2 2 ) ∪ {vβ(v � ) | v ∈ V 1 \ W 1 , v � ∈ W 1 , vv � ∈ E 1 } ∪ {β(v)β(v � ) | v, v � ∈ W 1 , vv � ∈ E 1 }; 3. For all v 1 , v � 1 ∈ V 1 \ W 1 , w 1 , w � 1 ∈ W 1 , v 2 , v � 2 ∈ V 2 , define l 3 (v 1 v � 1 ) = l 1 (v 1 v � 1 ), l 3 (v 1 β(w 1 )) = l 1 (v 1 w 1 ), l 3 (v 2 v � 2 ) = l 2 (v 2 v � 2 ), l 3 (β(w 1 )β(w � 1 )) = l 1 (w 1 w � 1 ); 4. F 3 = (F 1 \ W 1 ) ∪ β(F 1 ∩ W 1 ) ∪ F 2 ; 5. φ 03 | F 1 \W 1 = φ 01 | F 1 \W 1 , φ 03 • β = φ 01 | W 1 , φ 03 | F 2 \β(W 1 ) = φ 02 | F 2 \β(W 1 ) ; 6. P 3 = (P 1 \ W 1 ) ∪ β(P 1 ∩ W 1 ) ∪ (P 2 \ β(W 1 )); 7. Q 3 = (Q 1 \ W 1 ) ∪ Q 2 .
The combination of two linkages is prohibited in the following cases:

1. There exist

a 1 , b 1 ∈ F 1 ∩ W 1 such that β(a 1 ) = β(b 1
) and φ 01 (a 1 ) � = φ 01 (b 1 ) (two vertices are fixed at different places but should be attached to the same other vertex).

2. There exist

a 1 , b 1 ∈ W 1 such that a 1 b 1 ∈ E 1 , β(a 1 )β(b 1 ) ∈ E 2 , and l 1 (a 1 b 1 ) � = l 2 (β(a 1 )β(b 1 )

) (two edges of different lengths should join one couple of vertices).

3. There exist

a 1 ∈ V 1 and b 1 , c 1 ∈ W 1 , such that a 1 b 1 ∈ E 1 , a 1 c 1 ∈ E 1 , β(b 1 ) = β(c 1 ), and l 1 (a 1 b 1 ) � = l 1 (a 1 c 1 )

(again, two edges of different lengths should join one couple of vertices).

Example 2.25. Consider the two identical linkages L 1 and L 2 :

a 1 b 1 c 1 l 2 l 1 a 2 b 2 c 2 l 2 l 1
The inputs of L i are a i , b i and the output is c i .

To combine the two linkages, let

W 1 = {c 1 } and β(c 1 ) = a 2 . Then L 3 := L 1 ∪ β L 2 is the following linkage: a 1 b 1 a 2 b 2 c 2 l 2 l 1 l 2 l 1
The inputs of L 3 are a 1 , b 1 , b 2 and the output is c 2 .

We end this section with three facts whose proofs are straightforward. The first describes Conf M (L) when L is obtained as the combination of two linkages, the second one describes Reg M (L), while the third one establishes a link between the combination of functional linkages and the composition of functions.

Fact 2.26. Let L 1 , L 2 be two linkages, W 1 ⊆ V 1 , β : W 1 → V 2 , and L 3 = L 1 ∪ β L 2 be defined as in Definition 2.24. Then Conf M (L 3 ) = � φ 3 ∈ M V 3   ∃(φ 1 , φ 2 ) ∈ Conf M (L 1 ) × Conf M (L 2 ) φ 1 | V 1 \W 1 = φ 3 | V 1 \W 1 , φ 1 | W 1 = φ 3 | β(W 1 ) • β, φ 2 = φ 3 | V 2 � . Fact 2.27. Let L 1 , L 2 be two linkages, W 1 ⊆ V 1 , β : W 1 → P 2 , and L 3 = L 1 ∪ β L 2 .
Suppose that ψ 3 ∈ Conf P 3 M (L 3 ) satisfies both of the following properties:

1. ∃ψ 1 ∈ Reg P 1 M (L 1 ) ψ 1 | P 1 \W 1 = ψ 3 | P 1 \W 1 , ψ 1 | P 1 ∩W 1 = ψ 3 | β(P 1 ∩W 1 ) • β; 2. ∀φ ∈ p -1 3 (ψ 3 ) φ| P 2 ∈ Reg P 2 M (L 2 ).
Then

ψ 3 ∈ Reg P 3 M (L 3 ). Fact 2.28. Let L 1 , L 2 be two linkages with card(Q 1 ) = card(P 2 ). Assume that L 1 is a functional linkage for f 1 : Conf P 1 M (L 1 ) → M Q 1 and that L 2 is a functional linkage for f 2 : Conf P 2 M (L 2 ) → M Q 2 . Let W 1 = Q 1 , β : W 1 → P 2 a bijection, and L 3 = L 1 ∪ β L 2 . The bijection β induces a bijection β between M Q 1 and M P 2 . Then L 3 is functional for f 2 • β • f 1 | Conf P 3 M (L 3 ) .

Appendix: Linkages on any Riemannian manifold

The aim of this section is to prove Theorem 2.10. Consider a linkage L = (V, E, F, l, φ 0 , P, Q) in the Euclidean space E n as in the statement of the theorem: we may assume without loss of generality that L is a connected graph, that the sum of the lengths of the edges is smaller than 1, and that one of the vertices is fixed to 0, so that the configuration space of L is a subset of B V , where B is the unit ball of E n . We introduce C the set of all mappings φ : V → B such that φ| F = φ 0 (namely, those which map the fixed points to their assigned locations), and define the mapping

Φ : C → (R ≥0 ) E φ � -→ � (v, v � ) � → �φ(v � ) -φ(v � )� 2 � .
Then the configuration space of L in E n is Φ -1 (l 2 ). Making a small perturbation of l, we may assume by the Lemma of Sard that l 2 is a regular value of Φ. By assumption, this perturbation does not change Conf E n (L), up to diffeomorphism.

Let U be an open neighborhood of 0 in E n , equipped with a metric g, such that (U, g) is isometric to an open subset of the Riemannian manifold M, and denote by δ the associated distance on U . Applying a linear transformation to U , we may assume that g 0 (the metric g at 0) coincides with the canonical Euclidean scalar product on R n .

For a small enough r > 0, the mapping

Ψ : C × ((-r, r) \ {0}) → R E × R � φ λ � � -→ � ψ : (v, v � ) � → 1 λ 2 δ 2 (λφ(v), λφ(v � )) λ
� is well-defined, smooth, and may be extended smoothly to C × (-r, r) (apply Taylor's formula).

Then for all small enough λ ∈ R,

Ψ -1 � l 2 λ � = 1 λ Conf (U,g) (L λ ) × {λ},
where

L λ = (V, E, F, λl, λφ 0 , P, Q). Notice that Conf (U,g) (L λ ) is diffeomorphic to the configuration space of some linkage in M, since (U, g) is isometric to an open set of M.
The key to the proof is the following fact:

Fact 2.29. For all φ ∈ C, Ψ � C 0 � = � Φ(C) 0 � .
Proof. In this proof, for any open set W ∈ R n , we will write C 1 ([0, 1], W ) the set of C 1 paths which take their values in W .

Let φ ∈ C. For any small enough λ ∈ R, we have:

Ψ 1 � φ λ � • (v, v � ) = 1 λ 2 inf γ∈C 1 ([0,1],U ) γ(0)=λφ(v), γ(1)=λφ(v � ) �� 1 0 � g γ(t) (γ � (t), γ � (t))dt � 2 = inf γ∈C 1 ([0,1], 1 λ U ) γ(0)=φ(v), γ(1)=φ(v � ) �� 1 0 � g λγ(t) (γ � (t), γ � (t))dt � 2 .
Any C 1 path from [0,[START_REF] Laetitia | is flat[END_REF] to R n takes it values in 1 λ U for some λ > 0. Thus, taking the limit as λ → 0, we obtain:

Ψ 1 � φ 0 � • (v, v � ) = inf γ∈C 1 ([0,1],R n ) γ(0)=φ(v), γ(1)=φ(v � ) �� 1 0 � g 0 (γ � (t), γ � (t))dt � 2 = �φ(v � ) -φ(v)� 2 ,
since g 0 is the canonical Euclidean scalar product.

Fact 2.29 shows that

� l 2 0 � is a regular value of Ψ, and that Ψ -1 � l 2 0 � = Conf E n (L).
Hence, for any small enough λ ∈ R,

Conf (U,g) (L λ ) is diffeomorphic to Conf E n (L), which ends the proof. Chapter 3

Linkages in the Minkowski plane

The aim of this chapter is to prove Theorem 2.4.

Generalities on the Minkowski plane

Notation

The Minkowski plane M is R 2 equipped with the bilinear form ϕ

�� x t � , � x � t � �� = xx � -tt � . The pseudo-norm �•� is defined by �α� 2 = ϕ(α, α), and �α� ∈ R ≥0 ∪ iR ≥0 . The "distance" between α 1 and α 2 is defined by δ(α 1 , α 2 ) = �α 1 -α 2 �.
For α ∈ M, we write x α and t α the usual coordinates in R 2 , so that �α� 2 = x 2 αt 2 α . Sometimes, it will be more convenient to use lightlike coordinates, defined by y α = x α + t α and z α = x αt α , so that �α� 2 = y α z α .

We write I =

��

x t

� ∈ M     t = 0 � .

Intersection of two hyperbolae

In the Minkowski plane, hyperbolae play a central role (instead of circles in the Euclidean plane): for any α ∈ M and r 2 ∈ R, the hyperbola H(α, r) is the set of all γ ∈ M such that δ(α, γ) 2 = r 2 . Let α 0 , α 1 ∈ M and r 2 0 , r 2 1 ∈ R, and write d = �α 0α 1 �. Our aim in this section is to determine the cardinality of

I = H(α 0 , r 0 ) ∩ H(α 1 , r 1 ). Proposition 3.1. If α 0 � = α 1 and r 2 0 r 2 1 � = 0, we have card(I) ≤ 2.
Proof. We write y 0 = y α 0 and z 0 = z α 0 . We may assume α 1 = 0 and y 0 � = 0. Then, I is the set of the solutions of the system with unknown (y, z):

� yz = r 2 1 (y -y 0 )(z -z 0 ) = r 2 0 which is equivalent to � yz = r 2 1 y 0 z 2 -(y 0 z 0 + r 2 1 -r 2 0 )z + r 2 1 z 0 = 0
Thus, z is one of the roots of a polynomial of degree 2 and y is fully determined by z, so there are at most two solutions to the system.

Proposition 3.2. 1. If r 2 0 r 2 1 < 0 and d 2 � = 0, then card(I) = 2. Moreover, if d � is the distance between the two points of I, then d 2 d � 2 < 0. 2. If r 2 0 r 2 1 < 0 and d 2 = 0, then card(I) = 1.
3. If r 2 0 r 2 1 > 0 and r 2 0 d 2 < 0, then card(I) = 2. Proof. Examine the following figures.

r 2 0 r 2 1 < 0 and d 2 � = 0. r 2 0 r 2 1 < 0 and d 2 = 0. r 2 0 r 2 1 > 0 and r 2 0 d 2 < 0.

The case of equality in the triangle inequality

In the Minkowski plane, the triangle inequality is not always valid, but the equality case is the same as in the Euclidean plane. Proof. We have

(�α� + �β�) 2 = �α + β� 2 �α� 2 + �β� 2 + 2�α��β� = �α� 2 + �β� 2 + 2ϕ(α, β) ϕ(α, β) = �α��β�.
Therefore, the discriminant of the polynomial function

λ � -→ �β� 2 λ 2 + 2ϕ(α, β)λ + �α� 2
is zero. Thus, �α + λβ� 2 is either nonnegative for all λ or nonpositive for all λ. This means that α and β are not linearly independent.

The dual linkage

Let L 1 be a linkage in the Minkowski plane. We define the reflection

s : C → C a + ib � -→ b + ia and construct L 2 , the dual linkage of L 1 , by 1. V 2 = V 1 2. E 2 = E 1 3. F 2 = F 1 4. P 2 = P 1 5. Q 2 = Q 1 6. l 2 = s • l 1 7. φ 02 = s • φ 01 (with R 2 identified to C with the coordinates (x, t)). For all W ⊆ V 1 , this linkage satisfies Conf W M (L 2 ) = � s • φ   φ ∈ Conf W M (L 1 )
� ,

Reg P 2 M (L 2 ) = � s • φ   φ ∈ Reg W M (L 1 )
� . We let P = {a, b} and F = ∅. We assume l 1 � = 0 and l 2 � = 0. We translate Proposition 3.2 in terms of linkages.

Elementary linkages for geometric operations

Fact 3.4.

1. If l 2 1 and l 2 2 have different signs,

Conf P M (L) ⊇ Reg P M (L) ⊇ � ψ ∈ M P   �ψ(a) -ψ(b)� � = 0 � 2. If l 2
1 and l 2 2 have the same sign,

Conf P M (L) ⊇ Reg P M (L) ⊇ � ψ ∈ M P   �ψ(a) -ψ(b)� 2 • l 2 1 < 0 � 3. More generally, let ψ ∈ Conf P M (L). If the intersection H(ψ(a), l 2 ) ∩ H(ψ(b), l 1 ) contains exactly two elements, then ψ ∈ Reg P M (L).
Proof. The fact that the intersection H(ψ(a), l 2 ) ∩ H(ψ(b), l 1 ) contains exactly two elements implies that these are obtained from simple roots of a polynomial of degree 2 (see the proof of Proposition 3.1). Therefore, locally, the roots depend smoothly on the coefficients.

The rigidified square linkage

This linkage is well-known in the Euclidean plane. It is the usual solution to the problem of degenerate configurations of the square. It is very useful to notice that it does behave in the same way in the Minkowski plane.

We first explain why we need to rigidify square linkages. If one considers the ordinary square linkage (see the following figure), there are many realizations φ in which φ(a)φ(b)φ(c)φ(d) is not a parallelogram (we call these realizations degenerate realizations of the square). This linkage is called the rigidified square. The input set is P = {a, c}. We assume l � = 0. Proposition 3.5.

1. For all φ ∈ Conf M (L) we have

φ(b) -φ(a) = φ(c) -φ(d) (φ(a)φ(b)φ(c)φ(d) is an "affine parallelogram"). 2. For all φ ∈ Conf M (L) such that φ(b) � = φ(d) and φ(a) � = φ(c), we have φ| P ∈ Reg P M (L). In particular, Reg P M (L) contains � ψ ∈ M {a,c}   �ψ(a) -ψ(c)� • l 2 < 0 � . Proof. 1. Let φ ∈ Conf M (L).
From the equality case in the triangle inequality, we have φ(f ) = φ(a)+φ(d)

2 and φ(e) = φ(b)+φ(c) 2 . Case 1: φ(a) = φ(c). In this case, �φ(f ) -φ(c)� = �φ(f ) -φ(a)�. Therefore, �φ(c) -φ(e)� + �φ(f ) -φ(c)� = �φ(f ) -φ(e)�,
(b) = φ(d) (this is again the first case) or φ(b) = φ(a) + φ(c) -φ(d), i.e. φ(b) -φ(a) = φ(c) -φ(d).
2. This is a consequence of Fact 3.4. In the Euclidean plane, it is well-known that the Peaucellier linkage is a functional linkage for the inversion with respect to the circle

The Peaucellier inversor

C � 0, � |R 2 -r 2 | � , that is, the function α � -→ |R 2 -r 2 | �α� 2 α.
In the Minkowski plane, we will prove that it is essentially functional for inversion with respect to the hyperbola

H � 0, √ R 2 + r 2 �
. More precisely, it is functional for the function α � -→ -R 2 +r 2 �α� 2 α (in the version of the Peaucellier inversor which we choose, a "-" sign appears). Then, (y φ(b) , z φ(b) ) and (y φ(c) , z φ(c) ) are the two solutions of the following system with unknown (y, z):

� yz = -r 2 (y -y φ(e) )(z -z φ(e) ) = R 2 .
This system is equivalent to

� yz = -r 2 -y φ(e) z 2 + (y φ(e) z φ(e) -r 2 -R 2 )z + r 2 z φ(e) = 0.
We deduce that

z φ(b) + z φ(c) = z φ(e) - r 2 + R 2 y φ(e)
and similarly

y φ(b) + y φ(c) = y φ(e) - r 2 + R 2 z φ(e)
which gives the desired result, since Proof. We give a detailed proof in order to illustrate the use of Fact 2.27. This method is the key to many proofs concerning Reg P M (L). The Peaucellier inversor may be seen as the combination of the following linkages: L 1 : a robotic arm {a 1 , c 1 , e 1 } with one input e 1 and one fixed vertex a 1 , one edge {a 1 , c 1 } of length ir and one edge {c 1 , e 1 } of length R;

φ(d) = φ(b) + φ(c) -φ(e).
c 3 e 3 R R R R b 4 c 4 g 4 l il L 1 L 2 L 3 L 4
We combine the linkages in the following way (observe that the name of the vertices are chosen so that each β i preserves the letters and only changes indices):

1. Let W 1 = {c 1 , e 1 }. Let β 1 (c 1 ) = c 3 , β 1 (e 1 ) = e 3 . Let L 5 = L 1 ∪ β 1 L 3 . The input set of L 5 is P 5 = {e 3 , b 3 }. 2. Let W 2 = {b 2 , e 2 }. Let β 2 (b 2 ) = b 3 , β 2 (e 2 ) = e 3 . Let L 6 = L 2 ∪ β 2 L 5 . The input set of L 6 is P 6 = {e 3 }. 3. Let W 6 = {b 3 , c 3 }. Let β 6 (b 3 ) = b 4 , β 6 (c 3 ) = c 4 . Let L 7 = L 6 ∪ β 6 L 4 . The input set of L 7 is P 7 = {e 3 }.
The linkage L 7 is exactly the Peaucellier linkage. Let ψ ∈ Conf P 1 M (L 1 ) such that the intersection H(0, ir) ∩ H(ψ(e 1 ), R) has cardinality 2. Facts 3.4 and 2.23 show that ψ ∈ Reg P 1 M (L 1 ). We may naturally identify Conf R) has cardinality at least 2, but it is in fact exactly 2 from Proposition 3.1. Therefore,

{e 3 ,b 4 } M (L 7 ) with a subset C of Conf P 5 M (L 5 ) (identifying b 4 with b 3 ). Let us show, using Fact 2.27, that C is in fact a subset of Reg P 5 M (L 5 ). Let ψ ∈ C, and let φ ∈ Conf V 7 M (L 7 ) such that φ(e 3 ) = ψ(e 3 ) and φ(b 4 ) = ψ(b 3 ). Let ψ 1 ∈ M {e 1 } defined by ψ 1 (e 1 ) = ψ(e 3 ): since φ(b 4 ) � = φ(c 3 ), the intersection H(0, ir) ∩ H(ψ 1 (e 1 ),
ψ 1 ∈ Reg P 1 M (L 1 )
, so the first hypothesis of Fact 2.27 is satisfied. For the second hypothesis, we need to show that φ| P 3 ∈ Reg P 3 M (L 3 ). We know that φ(b 4 ) � = φ(c 4 ), and from Proposition 3.6, we also know that φ(e 3 ) � = φ(d 3 ). Therefore, Proposition 3.5 tells us that φ| P 3 ∈ Reg P 3 M (L 3 ). The two hypotheses of Fact 2.27 are satisfied, so C ⊆ Reg P 5 M (L 5 ). In the same way, one may show that Conf

{e 3 } M (L 7 ) ⊆ Reg P 6
M (L 6 ), and finally, that Conf

{e 3 } M (L 7 ) ⊆ Reg P 7 M (L 7 ), so Reg P M (L 7 ) = Conf P M (L 7 ).
Proposition 3.9. For all φ ∈ Conf M (L) we have the equivalence

φ(d) ∈ H �� 0 -1 � , i � ⇐⇒ y φ(e) -z φ(e) = -(R 2 + r 2 ).
Proof. Let φ ∈ Conf M (L). The following lines are equivalent:

φ(d) ∈ H �� 0 -1 � , i � (y φ(d) + 1)(z φ(d) -1) = -1 � - (R 2 + r 2 ) �φ(e)� 2 y φ(e) + 1 � � - (R 2 + r 2 ) �φ(e)� 2 z φ(e) -1 � = -1 y φ(e) -z φ(e) = -(R 2 + r 2 ).

3.2.4

The partial t 0 -line linkage

a d b c e g f R R R R ir ir l il i i R = r = 1 √ 2 ; l > 0; F = {a, f }; φ 0 (a) = � 5 t 0 + 1/2 � , φ 0 (f ) = � 5 t 0 -1/2 � ; P = {e}.
Proposition 3.10. The workspace of e, Conf P M (L), is contained in the line t = t 0 , but does not necessarily contain the whole line. More precisely

�� x t � ∈ M     t = t 0 , |x -5| > 1/2 � ⊆ Reg P M (L) = Conf P M (L) ⊆ �� x t � ∈ M     t = t 0 � .
Proof. We apply Fact 3.7, Propositions 3.8 and 3.9.

For example, this linkage with the choice t 0 = 0 will be called the partial (t 0 = 0)-line linkage.

The dual of this linkage (see Section 3.1.4) is called the partial x 0 -line linkage.

The t 0 -integer linkage

This linkage contains four vertices a, b, c, d which are restricted to move on I (the x-axis) using a partial t 0 -line linkages. More precisely, the t 0 -integer linkage is obtained as the combination of the linkage on the figure below with is combined with four partial t 0 -line linkages L i , i = 1 . . . We have

Conf {d} M (L) = �� -3 t 0 � , � -2 t 0 � , . . . , � 3 t 0 � , � 4 t 0 �� .
Moreover, Conf M (L) is a finite set so Reg P M (L) = Conf P M (L). We will use this linkage twice to construct more complex linkages. In Section 3.2.6, we could have used a simpler linkage with a configuration space of cardinality 2 instead of 8, but we need it to have cardinality at least 7 in Section 3.3.3.

The t 0 -line linkage

This linkage traces out the whole horizontal line t = t 0 : it contains a vertex e, the input, such that Conf {e} M (L) = {α ∈ M | t α = t 0 } . To construct it, the idea is to combine a partial t 0 -line linkage with a t 0 -integer linkage, as follows (to simplify the notations, we only give the construction of the (t 0 = 0)-line linkage). Let:

• L 1 be a (t 0 = 1
2 )-integer linkage;

• L 2 a (t 0 = -1 2 )-integer linkage;

• L 3 the combination (disjoint union) of the two linkages L 1 and L 2 ;

• L 4 a linkage similar to a partial t 0 -line linkage, with the only difference that

F 4 = ∅ instead of F 4 = {a 4 , f 4 }; • W 3 = {d 1 , d 2 } and β(d 1 ) = a 4 , β(d 2 ) = f 4 ; • L 5 = L 3 ∪ β L 4 . Since for any x ∈ R, we have either |x -5| > 1/2 or |x -7| > 1/2, we obtain as desired Conf {e 5 } M (L 5 ) = {α ∈ M | t α = t 0 } .
Using Fact 2.27, we also obtain Reg P M (L) = Conf P M (L). For future reference, we let a := a 1 , f := f 1 , e := e 4 . The dual of this linkage is called the x 0 -line linkage.

The horizontal parallelizer

This linkage has the input set P = {e 3 , e 4 }. It satisfies

Reg P M (L) = Conf P M (L) = � ψ ∈ M {e 3 ,e 4 }    t ψ(e 3 ) = t ψ(e 4 )
� .

Let:

• L 1 and L 2 be two (x 0 = 0)-line linkages;

• L 3 , L 4 two linkages similar to (t 0 = 0)-line linkages, but with F 3 , F 4 = ∅;

• L 5 the combination of L 1 and L 2 ;

• W 3 = {a 3 , f 3 }, β(a 3 ) = e 1 , β(f 3 ) = e 2 ,
and

L 6 = L 3 ∪ β L 5 ; • W 4 = {a 4 , f 4 }, β(a 4 ) = e 1 , β(f 4 ) = e 2 , and L 7 = L 4 ∪ β L 6 .
L 7 is the desired linkage.

For future reference, we let a := e 3 and b := e 4 . The dual of this linkage is called the vertical parallelizer.

The diagonal parallelizer

P = {a, b}, F = {g, f }, φ 0 (f ) = � 1 1 � , φ 0 (g) = � 0 0 � .
In this section, we use the lightlike coordinates y and z (see Section 3.1.1).

Proposition 3.11. We have

Reg P M (L) = Conf P M (L) = � ψ ∈ M P   y ψ(a) = y ψ(b) � .
Proof. The point is that for

α 1 , α 2 ∈ M such that y α 1 = y α 2 and α 1 � = α 2 , the intersection H(α 1 , 0) ∩ H(α 2 , 0
) is a straight line, more precisely:

H(α 1 , 0) ∩ H(α 2 , 0) = {γ | y γ = y α 1 } .

First, let us prove the inclusion Conf

P M (L) ⊆ � ψ ∈ M P   y ψ(a) = y ψ(b) � . For all φ ∈ Conf M (L), φ(c) ∈ H(φ(g), 0) ∩ H(φ(f ), 0) and z φ(f ) = z φ(g) , so z φ(c) = 0. Likewise, z φ(d) = 0. Since φ(e)
M (L) ⊇ � ψ ∈ M P   y ψ(a) = y ψ(b) � . Let ψ ∈ M {a,b} such that y ψ(a) = y ψ(b) . Construct φ ∈ Conf M (L) such that φ| {a,b} = ψ. Let φ(d) ∈ M such that z φ(d) = 0 and y φ(d) = y ψ(a) . Let φ(e) = φ(d) + � 1 -1 � and φ(c) = φ(d) + � -1 -1 � (in (x, t) coordinates). Then φ ∈ Conf M (L).
Finally, the coordinates of all vertices vary smoothly with respect to the coordinates of a and b, so Reg P M (L) = Conf P M (L).

Elementary linkages for algebraic operations 3.3.1 The average function linkage

The average function linkage is a linkage with the input set P = {a, b} and the output set Q = {c} which is a functional linkage for the function

f : I 2 → I (x 1 , x 2 ) � -→ x 1 + x 2 2 ,
and such that Reg P M (L) = Conf P M (L) = I P . Recall that by I, we mean the x-axis, and by "L is a functional linkage for f ", we mean that for all ψ ∈ Conf P M (L)

x ψ(c) = x ψ(a) + x ψ(b) 2 . a b c d e i i i i
The vertices a, b, c are restricted to move on the line I using (t 0 = 0)-line linkages: this means that the linkage in the figure above is combined with three (t 0 = 0)-line linkages. Likewise, the points e, d and c are restricted to have the same x coordinate using a vertical parallelizer. The square adbe is rigidified (thus, the actual average function linkage has much more than these 5 vertices, but many of them are not represented on the figure).

To see that this linkage is the desired functional linkage, first notice that φ(c) is the middle of the segment [φ(a), φ(b)] for all realization φ, because φ(a)φ(d)φ(b)φ(e) is a parallelogram. Moreover, the expression

t φ(d) = ± � 1 + � � x φ(b) -x φ(a) � � 2 2
shows that the coordinates of φ(d) (and similarly, φ(e)) depend on φ(a) and φ(b) in a differentiable way, so Reg P M (L) = I P .

The adder

The adder is a linkage with the input set P = {a 1 , b 1 } and the output set Q = {b 2 } which is a functional linkage for the function

f : I 2 → I (x 1 , x 2 ) � -→ x 1 + x 2 , with Reg P M (L) = Conf P M (L) = I P . It is constructed as L 1 ∪ β L 2 ,
where L 1 and L 2 are average function linkages, with

W 1 = {c 1 }, β(c 1 ) = c 2 , F 2 = {a 2 }, and φ 02 (a 2 ) = � 0 0
� .

Note that we may obtain a functional linkage for substraction by letting P = {b 2 , b 1 } and Q = {a 1 }. One may also construct (by induction on n) a functional linkage for x � → nx, where n is any integer, and (by switching the input and the output) a functional linkage for x � → 1 n x.

The square function linkage

The square function linkage is a linkage with the input set P = {a} and the output set Q = {b}: it is functional for the function

I → I x � -→ x 2 ,
with Reg P M (L) = Conf P M (L) = I P . To construct it, recall the algebraic trick described by Kapovich and Millson in [START_REF] Kapovich | Universality theorems for configuration spaces of planar linkages[END_REF]:

∀x ∈ R \ {-0.5, 0.5} x 2 = 0.25 + 1 1 x-0.5 -1 x+0.5
.

We have to find another trick to obtain a formula which works for every x ∈ R.

To do this, notice that for all x and x � in R we have the identity

x 2 = 2(x + x � ) 2 + 2(x � ) 2 -(x + 2x � ) 2 .
Thus the expression x 2 can be rewritten

2 � 0.25 + 1 1 x+x � -0.5 - 1 x+x � +0.5 � + 2 � 0.25 + 1 1 x � -0.5 -1 x � +0.5 � - � 0.25 + 1 1 x+2x � -0.5 - 1 x+2x � +0.5 � . (3.1)
Moreover, for all x ∈ R there exists an x � ∈ {-3, -2, . . . , 3, 4} such that

{x + x � , x + 2x � , x � } ∩ {-0.5, 0.5} = ∅.
Start with a (t 0 = 0)-integer linkage L 1 : think of the vertex d 1 as the number x � . Let new mobile vertex which will represent x and will be the input of the linkage (one does not add any new edge for now). Since Expression 3.1 is the composition of additions, subtractions and inversions, one may combine L 2 with linkages for addition, subtraction and inversion (for the inversion, use the Peaucellier inversor), in the spirit of Fact 2.28, so that the output of the new linkage L corresponds to Expression 3.1. This is the desired linkage.

The multiplier

The multiplier is a linkage with the input set P = {a, b} and the output set Q = {c} which is a functional linkage for the function

f : I 2 → I (x 1 , x 2 ) � -→ x 1 x 2 ,
such that Reg P M (L) = Conf P M (L) = I P . Simply construct the multiplier by combining square function linkages and adders, using the identity

∀x 1 , x 2 ∈ R x 1 x 2 = 1 4 � (x 1 + x 2 ) 2 -(x 1 -x 2 ) 2 � .

The polynomial linkage

Let f : R n → R m be a polynomial. We identify R with I.

The polynomial linkage is a functional linkage for the function f with card(P ) = n and Reg P M (L) = Conf P M (L) = I P . The polynomial linkage is obtained by combining adders and multipliers (use Fact 2.28). The coefficients are represented by fixed vertices.

Example To illustrate the general case, we give the following example: n = 2, m = 1, f (x, y) = 2x 3 y + π.

To construct a functional linkage for f , start with a linkage L consisting of two fixed Combine the new linkage with another multiplier: the combination mapping β sends c to one of the inputs and the output of L to the other one. The new linkage L is functional for (x, y) � -→ x 2 y.

Repeating this process once, we obtain a functional linkage for x 3 y, and then for 2x 3 y (using the vertex a).

Finally, combine the linkage L with an adder: the combination mapping β sends the output of L to one of the inputs, and b to the other one.

End of the proof of Theorem 2.4

Let n ∈ N. We are given A a semi-algebraic subset of (R 2 ) n , but we first assume that A is in fact an algebraic subset of (R 2 ) n , defined by a polynomial f : R 2n → R m (so that

A = f -1 (0)).
Take a polynomial linkage L for f . Name the elements of the input set: P = {a 1 , . . . , a 2n }. The output set Q has 2m elements.

The linkage L does not yet look like the desired linkage: since L has 2n inputs, the partial configuration space Conf P M (L) is a subset of (R 2 ) 2n (in fact, it is a subset of I 2n ), while A is a subset of (R 2 ) n (in particular, we are looking for a linkage with n inputs). To obtain Conf P M (L) = A ⊆ (R 2 ) n , we have to modify L in the following way.

1. With several (x 0 = 0)-line linkages and diagonal parallelizers, extend the linkage L to a new one with new vertices c 2 , c 4 , c 6 , . . . , c 2n such that for all realization φ and for all k ∈ {1, . . . , n} x φ(c 2k ) = 0;

y φ(c 2k ) = y φ(a 2k ) � i.e. x φ(c 2k ) + t φ(c 2k ) = x φ(a 2k ) + t φ(a 2k ) � .
2. With several vertical and horizontal parallelizers, extend this linkage to a new one with vertices d 2 , d 4 , d 6 , . . . , d 3n such that for all realization φ and for all k ∈ {1, . . . , n}

x φ(d 2k ) = x φ(a 2k-1 ) ; t φ(d 2k ) = t φ(c 2k ) .
Thus, for all realization φ and all k ∈ {1, . . . , n}, we have x φ(d 2k ) = x φ(a 2k-1 ) and t φ(d 2k ) = x φ(a 2k ) .

Let P = {d 2 , d 4 , . . . , d 2n }. Note that the input map p is a finite covering onto the simply connected set (R 2 ) n . Therefore, p is a trivial covering. The output set Q is unchanged.

x t a 1 a 2 c 2 d 2 Figure 3.3 -A partial realization of the four vertices a 1 , a 2 , c 2 , d 2 . We have x φ(d 2 ) = x φ(a 1 ) and t φ(d 2 ) = x φ(a 2 ) .
Fix the outputs to the origin: precisely, replace F by F ∪ Q and let

∀a ∈ Q φ 0 (a) = � 0 0 � .
We obtain as desired Reg P M (L) = Conf P M (L) = A. Finally, if A is any semi-algebraic set of (R 2 ) n , then A is the projection of an algebraic set B of (R 2 ) N for some N ≥ n. Construct the linkage L 1 such that Conf

Chapter 4

Linkages in the hyperbolic plane

The aim of this chapter is to prove Theorem 2.6.

Generalities on the hyperbolic plane

Definition 4.1. The Poincaré half-plane model is the half-plane:

�� x y � ∈ R 2     y > 0 � endowed with the metric: (dx) 2 + (dy) 2 y 2 .
This model is the one we will always use in this chapter.

The distance δ on H 2 is given by the formula:

δ �� x 1 y 1 � , � x 2 y 2 �� = arcosh � 1 + (x 2 -x 1 ) 2 + (y 2 -y 1 ) 2 2y 1 y 2 � .

Circles

In the Poincaré half-plane model, a hyperbolic circle with hyperbolic center α and hyperbolic radius R is in fact a Euclidean circle with center β and radius r, where

y β = y α cosh R, x β = x α and r = y α sinh R. Also notice that y β = � y 2 α + r 2 .

Some compact subsets of H 2

Since we work with linkages with compact configuration spaces, whereas H 2 is not compact, we need to introduce some compact subsets on which we will use our linkages. Fix a real parameter η > 1, and think of it as a very large number (the precise meaning of "large" will be given later). Let

I 0 := � α ∈ H 2   y α = 2, |x α | ≤ 10η � . J 0 := � α ∈ H 2   x α = 0, 2e -10η ≤ y α ≤ 2e 10η � . B 0 := � α ∈ H 2   |x α | ≤ 10η, 2e -10η ≤ y α ≤ 2e 10η � .
For any segment of positive length I, we write Î the line containing I. For example: We let P = {a, b}, l 1 > 0, l 2 > 0. We have:

Î0 := � α ∈ H 2   y α = 2 � . y = 0 I 0 J 0 B 0 4
Conf P H 2 (L) = � ψ ∈ (H 2 ) P   |l 1 -l 2 | ≤ δ(ψ(a), ψ(b)) ≤ l 1 + l 2 �
and Reg P H 2 (L) contains:

� ψ ∈ Conf P H 2 (L)   |l 1 -l 2 | < δ(ψ(a), ψ(b)) < l 1 + l 2 �
(recall that δ is the hyperbolic distance on H 2 ). We let F = {a}, P = {b}, Q = {c}. We require l � = r, t 1 � = t 2 , t 1 > r, t 2 > r.

y φ(b) = exp arcosh cosh r cosh D exp arcosh cosh l cosh D y φ(b) = cosh r + � (cosh r) 2 -(cosh D) 2 cosh l + � (cosh l) 2 -(cosh D) 2 y φ(c) = 1 � exp arcosh cosh r cosh D � � exp arcosh cosh l cosh D � y φ(c) = (cosh D) 2 � cosh r + � (cosh r) 2 -(cosh D) 2 � � cosh l + � (cosh l) 2 -(cosh D) 2 � .
Finally, we obtain as desired:

(y α -y φ(b) )(y α -y φ(c) ) = (cosh l) 2 (cosh r) 2 -1. General case. Let Φ : H 2 � → H 2 be an isometry such that Φ(φ(a)) = � 0 1 � , x Φ(µ) = 0
and y Φ(µ) ≤ 1. Let i be the inversion with respect to the circle with hyperbolic center � 0 1

� and hyperbolic radius arcosh cosh l cosh r . Then

φ(c) = Φ -1 • i • Φ(φ(b)), and
Φ -1 • i • Φ is the inversion with respect to the circle C.
We now study the workspace of the input b. Obviously, the input cannot be in the image of the lower half-plane by the inversion, because the output has to remain in the upper half-plane. Moreover, since the two edges (bf ) and (bc) have different lengths, the input cannot be a fixed point of the inversion. Let K be a compact set in H 2 \ (C ∪ D). Then there exists a choice of φ 0 (a), l, r, t 1 , t 2 such that the Peaucellier linkage with these lengths is functional for the inversion with respect to C, and such that K

⊆ Reg P H 2 (L) ⊆ Conf P H 2 (L). Proof. Let K � be a compact set in H 2 \ (C ∪ D) such that K ⊆ • K � (the interior of K � ). Let Q � , such that Q < Q � < R and K � ⊆ H 2 \ (C ∪ D � ), where D � is the closed disk of hyperbolic center α and hyperbolic radius Q � . Let Q �� > 0 such that K � ⊆ D �� , where D �� is the open disk of hyperbolic center α and hyperbolic radius Q �� . Define Φ : R ≥0 → R u → arcosh((cosh u)(cosh R)) -u. Since lim u→+∞ Φ(u) = log(cosh R) = Q, there exists u 0 ≥ Q �� such that Φ(u 0 ) ≤ Q � . Let φ 0 (a) = α, r = u 0 and l = Φ(u 0 ) + u 0 . Then, R = arcosh cosh l cosh r , l -r ≤ Q � and l + r ≥ Q �� .
Finally, choose t 1 and t 2 close enough to each other to have

K � ⊆ Conf P H 2 (L). Then K ⊆ • K � ⊆ Reg P H 2 (L).

The Euclidean line linkage

The aim of this linkage is to trace out any given Euclidean segment. More precisely, let Δ be a straight line and I ⊆ Δ a Euclidean segment: we construct a linkage L with one input such that

I ⊆ Reg P H 2 (L) ⊆ Conf P H 2 (L) ⊆ Δ. Let α ∈ R 2 , k 1 , k 2 > 0 such that:
1. Δ is outside the closed disk with hyperbolic center α and hyperbolic radius k 1 ; 2. I is contained in the open disk with hyperbolic center α and hyperbolic radius k 2 .

Let l and r such that l + r = k 2 . Choosing l and r sufficiently close to k 2 2 , we may also require arcosh cosh l cosh r ≤ k 1 . From Proposition 4.3, we deduce that there is a Peaucellier linkage L 1 such that I ⊆ Reg P H 2 (L) ⊆ Conf P H 2 (L). Let i be the inversion for which L is functional. Then i(Δ) is a circle contained in the half-plane (in which one point has been removed). Let L 2 be a circle linkage for this circle. Let W 1 = {c 1 } and β(c 1 ) = b 2 . Construct the combination L 3 = L 1 ∪ β L 2 . This linkage has the desired properties.

Rename the input: b := b 1 , and the fixed vertices: a := a 1 , c := a 2 . We also add an edge between the two fixed vertices a and c, of length δ(φ 03 (a), φ 03 (c)). This new edge will be useful for Sections 4.2.5 and 4.2.6.

The vertical parallelizer

The aim of this linkage is to force two vertices to have the same x coordinate. More precisely, it has two inputs a and b with:

� ψ ∈ (H 2 ) P   x ψ(a) = x ψ(b) � ∩ (B 0 ) P ⊆ Reg P H 2 (L) ⊆ Conf P H 2 (L) ⊆ � ψ ∈ (H 2 ) P   x ψ(a) = x ψ(b) �
(see the notations of Section 4.1.2).

To construct it, the idea is to allow two vertical Euclidean line linkages to move together horizontally.

Let L 1 and L 2 be two identical line linkages for the segment J 0 . Construct their disjoint union

L 3 = L 1 ∪ L 2 .
Let I be a horizontal segment of Euclidean length 20η centered at φ 01 (a 1 ), and I � a horizontal segment of Euclidean length 20η centered at φ 01 (c 1 ). Let L 4 and L 5 be line linkages for I and I � respectively, and construct their disjoint union

L 6 = L 4 ∪ L 5 . Change F 3 to ∅, let W 3 = {a 1 , c 1 , a 2 , c 2 }, β(a 1 ) = β(a 2 ) = b 4 , β(c 1 ) = β(c 2 ) = b 5 and L 7 = L 3 ∪ β L 6 .
Rename the inputs:

a := b 1 , b := b 2 .
This linkage has the desired properties.

The hyperbolic alignment linkage

The hyperbolic alignment linkage and the equidistance linkage are not needed to prove the differential universality (Theorem 2.7), but we will use them to prove the algebraic universality (Theorem 2.6).

The hyperbolic alignment linkage forces its three inputs a, b, c to be on the same hyperbolic line. Fix a real constant l > 0, then define

A = � ψ ∈ (H 2 ) P   ψ(a)
B = � ψ ∈ (H 2 ) P   0 < max(δ(ψ(a), ψ(b)), δ(ψ(b), ψ(c)), δ(ψ(a), ψ(c))) ≤ l � .
We want to construct a linkage such that

A ∩ B ⊆ Reg P H 2 (L) ⊆ Conf P H 2 (L) ⊆ A.
Take a vertical hyperbolic segment J of hyperbolic length l. Note that J is also a Euclidean segment, and that any hyperbolic segment of length l is the image of J by a global isometry of H 2 .

Take three identical Euclidean line linkages L 1 , L 2 , L 3 for J, with no fixed vertices (F 1 = F 2 = F 3 = ∅), and glue these three linkages together: let

W 1 = {a 1 , c 1 }, β(a 1 ) = a 2 , β(c 1 ) = c 2 , and L 4 = L 1 ∪ β L 2 . Next, let W 4 = {a 2 , c 2 }, β(a 2 ) = a 3 , β(c 2 ) = c

The equidistance linkage

The equidistance linkage forces an input a to be equidistant from the two other inputs d and e.

Fix two real constants k 1 > 0, k 2 > 0 and define

A = � ψ ∈ (H 2 ) P   δ(ψ(a), ψ(d)) = δ(ψ(a), ψ(e)) � and B = � ψ ∈ (H 2 ) P   δ(ψ(d), ψ(e)) ≥ k 1 , δ(ψ(a), ψ(d)) ≤ k 2 � .
We want to construct a linkage L such that

A ∩ B ⊆ Reg P H 2 (L) ⊆ Conf P H 2 (L) ⊆ A.
Start with the following linkage: Finally, choose t 1 , t 2 > r with t 1 � = t 2 and |t 1t 2 | sufficiently small.

Elementary linkages for algebraic operations

In this section, we describe linkages which are functional for algebraic operations such as addition or multiplication on real numbers. The real line is identified with Î0

= � α ∈ H 2   y α = 2 �
, which means that we will write simply x instead of (x, 2).

The symmetrizer

First version

The symmetrizer is a functional linkage for:

f : Conf P H 2 (L)(⊆ Î0 P ) → Î0 (x 1 , x 2 ) � -→ x 1 + x 2 2 . b d c f a r r t 1 t 2 Let P = {b, c}, Q = {a}, t 1 > r, t 2 > r, |t 1 -t 2 | = 1, r = 8η.
The vertices b and c are restricted to move on I 0 using two line linkages: this means that the linkage on the figure above is combined with two line linkages, which are not represented on the figure, with a combination map β such that β(b) and β(c) are the two inputs of the line linkages. The vertices a and d are restricted to move on the same vertical line using a vertical parallelizer.

For this linkage,

Conf P H 2 (L) = � ψ ∈ Î0 P    1 ≤ |ψ(b) -ψ(c)| ≤ 16η � and Reg P H 2 (L) contains � ψ ∈ Î0 P    1 < |ψ(b) -ψ(c)| < 16η � .
Modifying Conf P H 2 (L) Since we want the symmetrizer to handle input vertices b and c which are close to each other, or even equal, the first version is not suitable for our purpose. Notice that

x 1 + x 2 2 =
x 1 +8η 2

+ x 2 +(-8η) 2 2 .
Following this formula and the idea of Fact 2.28, take one symmetrizer L 1 , but add one input to the set of fixed vertices and set it to the coordinate � 8η 2

�

. Next, take a second symmetrizer L 2 , add one input to the set of fixed vertices and set it to

� -8η 2
� .
Finally, take a third symmetrizer L 3 and combine it with L 1 and L 2 , using a combination mapping β which sends the outputs of L 1 , L 2 to the inputs of L 3 . Thus, by combining three symmetrizers we get a new version of the symmetrizer L, which is functional for the same function, but such that Reg P H 2 (L) (⊆ Conf P H 2 (L)) contains:

� ψ ∈ Î0 P    |ψ(b)| < 7η, |ψ(c)| < 7η � .

The adder

Take a symmetrizer L. Fix the vertex b to 0, let P = {a} and Q = {c}. We obtain a functional linkage for x � -→ 2x, for which Reg P H 2 (L) contains:

� ψ ∈ Î0 P    |ψ(a)| < 3η � .
Combining this linkage with the symmetrizer, we get a functional linkage for

(x 1 , x 2 ) � -→ 2 x 1 + x 2 2 = x 1 + x 2 .
For this linkage, Reg P H 2 (L) contains:

� ψ ∈ Î0 P    |ψ(a)| < 3η, |ψ(b)| < 3η � .
This linkage is called the adder.

The opposite value linkage

Take a symmetrizer L. Fix the vertex a to 0, let P = {c} and Q = {b}. We obtain a functional linkage for x � -→ -x, for which Reg P H 2 (L) contains:

� ψ ∈ Î0 P    |ψ(c)| < 7η � .

The rational homothety linkage

Let n be an integer. Using n -1 adders, we get a functional linkage for x � -→ nx, for which Reg

P H 2 (L) contains � ψ ∈ Î0 P     |ψ(c)| < 3 n η � .
Switching the input and the output, we get a functional linkage for x � -→ x n , for which Reg P H 2 (L) contains

� ψ ∈ Î0 P    |ψ(c)| < 3η � .

The square function linkage

With the input set P = {a} and the output set Q = {b}, it is a functional linkage for the function:

f : Conf P H 2 (L)(⊆ Î0 ) → Î0 x � -→ x 2 ,
f : Conf P H 2 (L)(⊆ Î0 ) → Î0 x � -→ 1 x .
Now, as in the Minkowski case, we use the algebraic trick first described by Kapovich and Millson [START_REF] Kapovich | Universality theorems for configuration spaces of planar linkages[END_REF]:

∀x ∈ R \ {-0.5, 0.5} x 2 = 0.25 + 1 1 x-0.5 -1 x+0.5
, Thus, the desired linkage is obtained by composition of the previous linkages.

The multiplier

A first version

The multiplier is a linkage with P = {a, b} and Q = {c}, which is a functional linkage for the function:

f : Conf P H 2 (L)(⊆ Î0 P ) → Î0 (x 1 , x 2 ) � -→ x 1 x 2 such that Conf P H 2 (L) ⊆ Î0 P and Reg P H 2 (L) contains a neighborhood U of � 0 2 � in Î0 P .
We simply construct it by combining square function linkages and adders, using the identity:

∀x 1 , x 2 ∈ R x 1 x 2 = 1 4 ((x 1 + x 2 ) 2 -(x 1 -x 2 ) 2 )
.

Modifying Conf P H 2 (L)
We are now going to construct a multiplier such that Conf P H 2 (L) ⊆ Î0 P and Reg P H 2 (L) contains

�� x 2 �     x ∈ [-η, η] � P .
Let n be an integer such that

�� x 2 �     x ∈ [-η n , η n ]
� P ⊆ U (where U is defined in Section 4.3.6). Using two rational homothety linkages and one multiplier (first version), and the formula:

∀x 1 , x 2 ∈ R x 1 x 2 = n 2 � x 1 n � � x 2 n �
we obtain the desired linkage.

The polynomial linkage

Let f : R n → R m be a polynomial of degree d ≥ 1 and coefficients in [-K, K] with 1 ≤ K ≤ η. We still identify R with Î0 . Our aim is to construct a functional linkage for f

| Conf P H 2 (L) , with Conf P H 2 (L) ⊆ Î0 P and Reg P H 2 (L) containing U K,d,n := [-M K,d,n , M K,d,n ] P , where M K,d,n = 1 K(d+1) n η 1/d .
It is obtained by combining adders and multipliers. The coefficients are represented by fixed vertices.

Remark. At this stage, it would be possible to fix the outputs of the polynomial linkage to � 0 2 � to prove directly Theorem 2.7. However, the proof of Theorem 2.6 is more complicated, since we need the input vertices to move outside the line Î0 , as it will be explained in the next section.

End of the proof of Theorem 2.6

Let A be a compact semi-algebraic subset of (H 2 ) n . First, we assume that A is a compact algebraic subset of (H 2 ) n .

We want to construct a linkage with P = {d 2 , d 4 , . . . , d 2n } such that Conf P H 2 (L) = A. The idea is to identify each point α of the Poincaré half-plane with three coordinates X 1 α , X 2 α , X 3 α , defined by:

∀i ∈ {1, 2, 3} X i α = δ �� i 2 � , α � . Since � 1 2 � , � 2 2 � , � 3 2
� are not aligned, these three coordinates characterize the point

α. Let f : (R 2 ) n = R 2n → R m be a polynomial function (of degree d) with coefficients in [-1, 1] such that A = f -1 (0).
We may assume that A is contained in the set

V := ((-M 1,d,2n , M 1,d,2n ) × (3, M 1,d,2n )) n
(see Section 4.3.7 for the definition of M 1,d,2n ). If it is not, choose an isometry Φ of H 2 such that Φ(A) is contained in this set (for a large enough η), construct the desired linkage, and then replace φ 0 by Φ -1 • φ 0 .

If necessary, increase η (but do not change the definition of V by doing so) so that

M 100,2,2 ≥ max    |x|       x ∈ R, i ∈ {1, 2, 3}, δ �� x 2 � , � i 2 �� ≤ max (α1,...,αn)∈V k∈{1,...,n} δ � α k , � i 2 ��    .
We which are restricted to move on Î0 using line linkages.

Combine the linkage with equidistance linkages (with parameters k 1 = 1 and k 2 = M 1,d,2n + 4) so that for all φ ∈ Conf H 2 (L), all k ∈ {1, . . . , n} and all i ∈ {1, 2, 3}:

X i φ(b i 2k ) = X i φ(d 2k ) .
Then, use polynomial linkages so that for all φ ∈ Conf H 2 (L), all k ∈ {1, . . . , n} and all i ∈ {1, 2, 3}:

x φ(c i 2k ) = x φ(a 2k ) • (x φ(b i 2k ) -i) 2 and x φ(c i 2k ) = 2 • ((x φ(a 2k-1 ) -i) 2 + (x φ(a 2k ) -2) 2
). Thus we have for all i and k:

x φ(a 2k ) • (x φ(b i 2k ) -i) 2 = 2 • ((x φ(a 2k-1 ) -i) 2 + (x φ(a 2k ) -2) 2 ) arcosh � 1 + (x φ(b i 2k ) -i) 2 2 • 2 � = arcosh � 1 + (x φ(a 2k-1 ) -i) 2 + (x φ(a 2k ) -2) 2 2 • x φ(a 2k ) � δ � φ(b i 2k ), � i 2 �� = δ �� x φ(a 2k-1 ) x φ(a 2k ) � , � i 2 �� X i φ(b i 2k ) = X i � x φ(a 2k-1 )
x φ(a 2k )

� X i φ(d 2k ) = X i � x φ(a 2k-1 )
x φ(a 2k )

� .

Thus:

φ(d 2k ) = � x φ(a 2k-1 )
x φ(a 2k )

� .

Add vertices e 1 , . . . , e m and use a polynomial linkage so that for all φ ∈ Conf H 2 (L):

f (x φ(a 1 ) , . . . , x φ(a 2n ) ) = (x φ(e 1 ) , . . . , x φ(em) ). Now, notice that p| π -1 (V) is a smooth finite covering onto V, which is necessarily trivial since V is simply connected.

To finish the construction, fix the vertices e 1 , . . . , e m to the point � 0 2

� .

Thus, Reg

{a 1 ,...,a 2n } H 2 (L) = Conf {a 1 ,...,a 2n } H 2 (L) = A ⊆ R 2n = Î0 ,
and finally:

Reg P H 2 (L) = Conf P H 2 (L) = A ⊆ (H 2 ) n ⊆ (R 2 ) n . Moreover, the restriction map Conf H 2 (L) → Conf P H 2 (L)
is a smooth finite covering, which is trivial as the restriction of a trivial covering.

If A is only a compact semi-algebraic subset of (H 2 ) n , we know from Proposition 2.14 that A is the projection onto the first coordinates of a compact algebraic set B. Apply the above construction to B and remove some vertices from the input set to obtain Conf P H 2 (L) = A, which ends the proof of Theorem 2.6.

Chapter 5

Linkages in the sphere

The aim of this chapter is to prove Theorem 2.9. In the first three sections, we focus on the two-dimensional sphere, while higher dimensions are studied in the last section.

The sphere S 2 will be considered as the unit sphere of R 3 . Thus, a point α ∈ S 2 is denoted by three coordinates x α , y α , z α . A standard computation gives:

Elementary linkages for geometric operations

Conf P S 2 (L) = � ψ ∈ (S 2 ) P   |l 1 -l 2 | ≤ δ(ψ(a), ψ(b)) ≤ min(l 1 + l 2 , 2π -(l 1 + l 2 )) � .

The great circle linkage

One vertex a is linked to k other vertices v 1 , . . . , v k , by edges of length π/2.

v k • • • v 3 v 2 v 1 a π/2 π/2 π/2 π/2 P = {v 1 , . . . , v k }, Q = ∅, F = ∅.
This linkage forces v 1 , . . . , v k to be on the same great circle of the sphere:

Conf P S 2 (L) = � ψ ∈ (S 2 ) P   ∃f ∈ (R 3 ) * ∀i ∈ {1, . . . , k} f (ψ(v k )) = 0 � .
The fixed great circle linkage is a variant in which a ∈ F . Then Conf P S 2 (L) = (S 2 ∩ a ⊥ ) P .

The symmetrizer

The symmetrizer is a functional linkage for symmetry with respect to a great circle (i.e. orthogonal symmetry with respect to a plane P in R 3 ). It is the key to the construction of several other linkages, but it is also the most difficult to obtain.

First version

Here is a first attempt, which we shall call L 1 . All the edges have length π/2, except when another length is indicated.

Let P = {a, b}, Q = {c}, and F = ∅. We want a to be the unit normal vector to the (linear) plane P of symmetry, and b to be the point to which we want to apply the symmetry. The result of the symmetry is c. Proposition 5.1. Fix some ψ ∈ (S 2 ) P . Let α ∈ S 2 be symmetric to ψ(b) with respect to ψ(a) ⊥ . Then:

� φ(c)   φ ∈ p -1 (ψ) � = {α, -α}.
Proof. Assume that φ ∈ Conf P S 2 (L 1 ) is such that φ| P = ψ. Applying a rotation to the sphere if necessary, we may assume that ψ(a) = (0, 0, 1) and y ψ(b) = 0. Let φ ∈ Conf S 2 (L 1 ). If x ψ(b) = 0, we may also assume up to rotation that y φ(d) = 0. If x ψ(b) � = 0, then φ(g) ⊥ contains the two distinct points ψ(a) and ψ(b), so φ(g) ∈ {±(0, 1, 0)}. Applying a symmetry with respect to φ(g) ⊥ if necessary, we may assume that φ(g) = (0, 1, 0). But φ(d) ∈ φ(g) ⊥ ∩ φ(a) ⊥ , so that φ(d) ∈ {±(1, 0, 0)}. Therefore, whether or not x ψ(b) = 0, we may assume φ(d) ∈ {±(1, 0, 0)} and φ(g) = (0, 1, 0). Hence,

φ(i) = (0, 1/2, √ 3/2) and φ(j) = (0, 1/2, - √ 3/2). Since φ(k) is on the line φ(i) ⊥ ∩ φ(b) ⊥ , it has two possible (opposite) values. φ(k) � ∈ Rφ(a) because φ(k) ∈ φ(i) ⊥ and φ(a) � ∈ φ(i) ⊥ .
Since m is on the line φ(k) ⊥ ∩ φ(a) ⊥ , it has two possible opposite values. Since l is on the line φ(j) ⊥ ∩ φ(m) ⊥ , φ(l) has two possible opposite values.

φ(l) � ∈ Rφ(g) because φ(l) ∈ φ(j) ⊥ and φ(g) � ∈ φ(j) ⊥ .

Since c is on the line φ(l) ⊥ ∩ φ(g) ⊥ , φ(c) has two possible opposite values.

Note that the construction of φ described above really provides a realization of the linkage, which proves that Conf P S 2 (L 1 ) is the whole (S 2 ) P . To see that one of the possible values of φ(c) is symmetric to φ(b) with respect to φ(a) ⊥ , use the symmetries of the abstract linkage: take φ ∈ Conf {a,b,d,g,i,k,m} S 2 (L 1 ), and extend φ to V (the set of all vertices) by letting φ(c) be symmetric to φ(b) with respect to φ(a) ⊥ , φ(l) symmetric to φ(k) with respect to φ(a) ⊥ , and φ(j) symmetric to φ(i) with respect to φ(a) ⊥ . Then it is clear that φ ∈ Conf S 2 (L 1 ).

In order to avoid the configurations in which φ(c) is not symmetric to φ(b) with respect to φ(a) ⊥ , we introduce a second version of the symmetrizer.

Second version

Here is a different version of the symmetrizer, L 2 : All the edges have length π/2, except when another length is indicated; P = {a, b}, Q = {c}, and F = ∅. As before, we want c to be symmetric to b with respect to a ⊥ . Proposition 5.2.

1. For all ψ ∈ (S 2 ) P , there exists φ ∈ p -1 (ψ) such that φ(c) is symmetric to φ(b) with respect to φ(a) ⊥ . L 2 is not a functional linkage for symmetry. There is a possible degenerate configuration which seems difficult to avoid: for any position of the inputs ψ

There does not exist

φ ∈ Conf S 2 (L 2 ) such that -φ(c)
∈ Conf P S 2 (L 2 ), there is a φ ∈ Conf S 2 (L 2 ) such that φ(b) = φ(d).
This problem is very related to the problem of the degenerate configurations of the parallelogram, which Kempe did not see when he wrote his original proof. The solution to this problem in the plane is the rigidification of the parallelogram, but the usual rigidification does not work in the sphere.

Gluing the two versions

We now have two linkages, L 1 and L 2 , which are almost functional linkages for symmetry, and have different degenerate configurations.

We glue them together: let

W 1 = {a 1 , b 1 , c 1 } and β(a 1 ) = a 2 , β(b 1 ) = b 2 , β(c 1 ) = c 2 , and L = L 1 ∪ β L 2 .
We rename some vertices for future reference:

a := a 2 , b := b 2 , c := c 2 , d := d 2 , g := g 2 .
Proposition 5.3.

1. Conf P S 2 (L) = (S 2 ) P .

2. L is a functional linkage for symmetry: for all φ ∈ Conf S 2 (L), φ(c) is symmetric to φ(b) with respect to φ(a) ⊥ .

Proof. This is an immediate consequence of propositions 5.1 and 5.2.

The parallelizer

The parallelizer has three inputs b, c, h, such that

Conf P S 2 (L) = � ψ ∈ (S 2 ) P   δ(ψ(h), ψ(b)) = δ(ψ(h), ψ(c)) � .
Notice that the equality δ(ψ(h), ψ(b

)) = δ(ψ(h), ψ(c)) is equivalent to (ψ(a)|φ(h)) = (ψ(b)|ψ(h))
, where (•|•) denotes the scalar product in R 3 . Therefore, for any linear form f : R 3 → R, there exists α ∈ S 2 such that any realization φ of the parallelizer with the vertex h fixed at α satisfies:

f (φ(a)) = f (φ(b)).
To construct the parallelizer, we use the following characterization: δ(ψ(h), ψ(b)) = δ(ψ(h), ψ(c)) if and only if there exists a linear plane P containing ψ(h) such that ψ(b) is the reflection of ψ(c) with respect to P.

Start with a symmetrizer L 1 and consider the following linkage L 2 :

a 2 h 2 π/2 P 2 = {a 2 , h 2 }, Q 2 = ∅, F 2 = ∅. Then let W 1 = {a 1 }, β(a 1 ) = a 2 , and L = L 1 ∪ β L 2 .
Change the input set of L so that P = {b 1 , c 1 , h 2 }, and rename the inputs: b = b 1 , c = c 1 , h = h 2 , which ends the construction. Start with the following linkage L 1 :

Elementary linkages for algebraic operations

a 1 b 1 c 1 arccos λ arcsin λ π/2 P 1 = {a 1 }, Q 1 = {b 1 }, F 1 = {c 1 }, φ 01 (c 1 ) = (0, 0, 1).
This linkage is functional for a homothety from the equator to the (smaller) circle of latitude arccos λ: thus for all φ ∈ Conf S 2 (L 1 ), y φ(b 1 ) = λy φ(a 1 ) , and Conf P 1 S 2 (L 1 ) = S 2 ∩ (Oxy) However, z φ(b 1 ) � = 0 so we need to improve the construction. Let L 2 be a parallelizer for the linear form f (x, y, z) = y and L 3 a parallelizer for g(x, y, z) = z.

Let W 1 = {b 1 }, β 1 (b 1 ) = b 2 , and L 4 = L 1 ∪ β 1 L 2 . Let W 4 = {a 1 , c 2 }, β 4 (a 1 ) = b 3 , β 4 (c 2 ) = c 3 , and L 5 = L 4 ∪ β 4 L 3 .
We get: Conf P 1 S 2 (L 1 ) = S 2 ∩ (Oxy), and for all φ ∈ Conf S 2 (L), y φ(c 3 ) = λy φ(a 1 ) , z φ(c 3 ) = 0.

Finally, rename the two vertices: a = a 1 and b = c 3 . There are several steps to construct such a linkage L. Then for all φ ∈ Conf S 2 (L) we have y φ(c) = y φ(a) + y φ(b) , as desired. Let P = {a, b} and Q = {c}.

The adder

We have

Conf P S 2 (L) = � ψ ∈ (S 2 ∩ (Oxy)) P   y φ(a) + y φ(b) ∈ [-1, 1] � .

The multiplier

Identify the plane (Oxy) with the complex plane: to a point (x, y, 0) ∈ R 3 , associate the complex number ζ (x,y,0) = x + iy. We want to construct a functional linkage which takes two complex numbers and returns their product. Since we work in the sphere, we only need to multiply complex numbers α 1 and α 2 in the unit circle. This corresponds to adding the arguments. We split this operation into two steps:

1. Compute arg(α 1 )+arg(α 2 ) 2 mod π;

Double the argument.

The following linkage L 1 will be the basis of the construction:

h 1 g 1 d 1 a 1 b 1 π/2 π/2 π/2 π/2 π/2 P 1 = {a 1 , b 1 }, Q = ∅, F 1 = {g 1 , h 1 }, φ 01 (g 1 ) =
(1, 0, 0), φ 01 (h 1 ) = (0, 0, 1). We have Conf P 1 S 2 (L 1 ) = (S 2 ) P 1 . Take two copies L 2 and L 3 of the symmetrizer. Let

W 1 = {a 1 , b 1 }, β 1 (a 1 ) = a 2 , β 1 (b 1 ) = b 2 , and L 5 = L 1 ∪ β 1 L 2 . Then let W 5 = {a 2 , g 1 }, β 5 (a 2 ) = a 3 , β 5 (g 1 ) = b 3 , L 4 = L 5 ∪ β 5 L 3 . We write a 4 := a 3 , b 4 := b 2 , c 4 := c 2 , d 4 := d 1 , f 4 := c 3 , g 4 := b 3 .
Now for all φ ∈ Conf S 2 (L 4 ), φ(c 4 ) is symmetric to φ(b 4 ) with respect to φ(a 4 ) ⊥ , and φ(f 4 ) is symmetric to φ(g 4 ) with respect to φ(a 4 ) ⊥ . In other words,

arg ζ φ(d 4 ) = arg(ζ φ(b 4 ) ) + arg(ζ φ(c 4 ) ) 2 mod π and arg ζ φ(f 4 ) = 2 arg ζ φ(d 4 ) mod 2π.
Let L 6 be the linkage L 4 with the input set P 6 = {b 4 , c 4 }. We have Conf P 6 S 2 (L 6 ) = (S 2 ) P 6 .

Taking Q 6 = {f 4 }, L 6 becomes a functional linkage for multiplication.

The polynomial linkage

Let f : R n → R m be a polynomial. Our aim is now to construct a linkage with n inputs a 1 , . . . , a n , such that:

Conf P S 2 (L) = � ψ ∈ (S 2 ∩ (Oxy)) P   f (y ψ(a 1 ) , . . . , y ψ(an) ) = 0 � .
Let us assume first that m = 1.

Recall that we write ζ ψ(a k ) = x ψ(a k ) + iy ψ(a k ) . We can also write:

y ψ(a k ) = ζ ψ(a k ) -ζ ψ(a k ) 2i .
Thus, there exists a polynomial g : C 2n → C such that for all ψ ∈ (S 2 ∩ (Oxy)) P :

g(ζ ψ(a 1 ) , ζ ψ(a 1 ) , . . . , ζ ψ(an) , ζ ψ(an) ) = f (y ψ(a 1 ) , . . . , y ψ(an) ).

We write

g = r � j=1 g j
where each g j is a monomial:

g j (ζ ψ(a1) , ζ ψ(a1) , . . . , ζ ψ(an) , ζ ψ(an) ) = λ j � j (ζ ψ(a1) ) γj,1 (ζ ψ(a1) ) γj,2 • • • (ζ ψ(an) ) γj,2n-1 (ζ ψ(an) ) γj,2n
with � j ∈ {1, i, -1, -i} and λ j a positive real number.

Observe that without changing the locus � ψ ∈ (S 2 ∩ (Oxy)) P   f (y ψ(a 1 ) , . . . , y ψ(an) ) = 0 � , one may assume λ j < λ 0 for all j, where λ 0 is arbitrary in (0, 1) (if necessary, multiply f by a small constant).

We are now ready to construct the linkage. Start with a fixed great circle linkage L which forces all the a k to move in the plane (Oxy). Use symmetrizers to extend L to a new linkage with vertices a � k such that for all k ∈ {1, . . . , n},

ζ φ(a � k ) = ζ φ(a k ) .
For each j ∈ {1, . . . , r}:

1. Use multipliers to extend L to a new linkage with a vertex c j such that for all φ ∈ Conf S 2 (L),

ζ φ(c j ) = (ζ ψ(a 1 ) ) γ j,1 (ζ ψ(a 1 ) ) γ j,2 • • • (ζ ψ(an) ) γ j,2n-1 (ζ ψ(an) ) γ j,2n .
2. Use a multiplier to extend the linkage to a new one with a vertex d j such that:

ζ φ(d j ) = i� j ζ φ(c j ) .
3. Use a homothety linkage to extend the linkage to a new one with a vertex b j such that: y φ(b j ) = λ j y φ(d j ) .

Thus we have for all φ ∈ Conf S 2 (L):

y φ(b j ) = �(ig j (ζ ψ(a 1 ) , ζ ψ(a 1 ) , . . . , ζ ψ(an) , ζ ψ(an) )).
Then, use several adders to extend the linkage to a new one, still called L, with a vertex c such that for all φ ∈ Conf S 2 (L):

y φ(c) = r � j=1 y φ(b j ) . Thus y φ(c) = Im(ig(ζ ψ(a 1 ) , ζ ψ(a 1 ) , . . . , ζ ψ(an) , ζ ψ(an) )),
which means that y φ(c) = f (y ψ(a 1 ) , . . . , y ψ(an) ).

Choose λ 0 so small that all the steps of the computation remain in [-1, 1]. Then:

Conf P S 2 (L) = (S 2 ∩ (Oxy)) P . Finally, if m ≥ 2, just write f = (f 1 , . . . , f m ) and use m linkages like above.

End of the proof of Theorem 2.9 for d = 2

In this section, we prove the algebraic universality in S 2 .

First, we assume that A is an algebraic subset of (S 2 ) n . Let f

: (R 3 ) n = R 3n → R m be a polynomial function such that A = f -1 (0).
1. Take a polynomial linkage L with inputs a 1 , . . . , a 3n such that: x φ(b 3k-2 ) = y φ(a 3k-2 ) .

3. With several symmetrizers, extend this linkage to a new one with vertices c 3 , c 6 , c 9 , . . . , c 3n such that for all φ ∈ Conf S 2 (L) and for all k ∈ {1, . . . , n}:

z φ(c 3k ) = y φ(a 3k ) .
4. With several parallelizers, extend this linkage to a new one with vertices d 3 , d 6 , d 9 , . . . , d 3n such that for all φ ∈ Conf S 2 (L) and for all k ∈ {1, . . . , n}:

x φ(d 3k ) = x φ(b 3k-2 ) ; y φ(d 3k ) = y φ(a 3k-1 ) ; z φ(d 3k ) = z φ(c 3k ) .
Now, let P = {d 3 , d 6 , . . . , d 3n }. We have:

Conf P S 2 (L) = f -1 (0) = A.
If A is only a compact semi-algebraic subset of (S 2 ) n , we know from Proposition 2.15 that A is the projection onto the first coordinates of an algebraic subset B of the sphere: apply the above construction to B and remove some vertices from the input set to obtain Conf P S 2 (L) = A; thus, Theorem 2.9 is proved.

Higher dimensions

In this section, we fix a number d ≥ 2 and consider realizations in the sphere S d .

The 3-plane linkage

This linkage forces several points to move in the same (linear) 3-plane. It is to be compared with the "great circle linkage" described in section 5.1.2, which forces several points to move in the same (linear) 2-plane.

There are k inputs v 1 , . . . , v k , and d -2 other vertices a 1 , . . . , a d-2 . For all i, j ∈ {1, . . . , d -2}, there is an edge a i a j of length π/2. For all i ∈ {1, . . . , d -2} and l ∈ {1, . . . , k}, there is an edge a i v l of length π/2.

Here is an example with d = 5 and k = 3.

a 1 a 2 a 3 v 1 v 2 v 3
Proposition 5.4. We have Conf P S d (L) = E, where

E = � ψ ∈ (S d ) P    ∃F subspace of R d+1 , dim F = 3, ∀i ∈ {1, . . . , k} ψ(v k ) ∈ F � .
Proof. First, we prove that Conf

P S d (L) ⊆ E. Let ψ ∈ Conf P S d (L) and φ ∈ p -1 (ψ). Let F = � 1≤i≤d-2 φ(a i ) ⊥ .
We know that {φ(a 1 ), . . . , φ(a d-2 )} is an orthonormal set, so dim F = 3. Moreover, for all l ∈ {1, . . . , k}, ψ(v l ) ∈ F. Now, we prove that E ⊆ Conf P S d (L). Let ψ ∈ E. Let F be a subspace of R d+1 with dim F = 3 containing ψ(v l ) for l ∈ {1, . . . , k}. Construct φ ∈ (S d ) P by letting {φ(a 1 ), . . . , φ(a d-2 )} ⊆ F ⊥ be an orthonormal set and let φ| P = ψ. Then φ ∈ Conf S d (L) so ψ ∈ Conf P S d (L).

The fixed 3-plane linkage is a variant in which a 1 , . . . , a d-2 ∈ F (namely, they are fixed vertices). Then there exists F a subspace of R d+1 with dim F = 3 and

Conf P S d (L) = � ψ ∈ (S 2 ) P   ∀i ∈ {1, . . . , k} ψ(v k ) ∈ F � .

The d-dimensional symmetrizer

Like in the 2-dimensional case, the d-dimensional symmetrizer has two inputs a and b, and one output c. It is a functional linkage for symmetry: for all φ ∈ Conf S d (L), φ(c) is symmetric to φ(b) with respect to φ(a) ⊥ . The idea is that the symmetry takes place in a 3-plane containing φ(a), φ(b) and φ(c).

Let L 1 be a classical symmetrizer and L 2 be a 3-plane linkage, with

k = card(V 1 ). Let W 1 = V 1 , β a bijection between V 1 and {v 1 , . . . , v k } (⊆ V 2 ), and L = L 1 ∪ β L 2 .
Letting a := β(a 1 ), b := β(b 1 ), c := β(c 1 ), we obtain as desired:

Conf S d (L) = � φ ∈ (S d ) V    φ(c) is symmetric to φ(b) with respect to φ(a) ⊥ � .

The d-dimensional parallelizer

Like in the 2-dimensional case, the d-dimensional parallelizer forces two points to have the same scalar product with a third one. We restrict the vertices of a classical parallelizer to move on a 3-plane containing its three inputs. Let L 1 be a classical parallelizer and L 2 be a 3-plane linkage, with

k = card(V 1 ). Let W 1 = V 1 , β a bijection between V 1 and {v 1 , . . . , v k } (⊆ V 2 ), and L = L 1 ∪ β L 2 . Let h := β(h 1 ), b := β(b 1 ), c := β(c 1
). Then we obtain as desired:

Conf P S d (L) = � ψ ∈ (S d ) P    δ(ψ(h), ψ(b)) = δ(ψ(h), ψ(c))
� .

5.4.4

End of the proof of Theorem 2.9 for d ≥ 2

Here, we prove the algebraic universality in S d . The proof is similar to the case d = 2. There are only two differences.

1. The polynomial linkage L is attached to a fixed 3-plane linkage.

2. We use d-dimensional symmetrizers and d-dimensional parallelizers.

Part II

Anosov geodesic flows, billiards and linkages

Chapter 6

Anosov geodesic flows and dispersing billiards

Introduction

The aim of this chapter is to highlight the similarities between two uniformly hyperbolic dynamical systems: geodesic flows on negatively curved manifolds and billiard flows on negatively curved billiards. A significant difference between these two dynamical systems is that geodesic flows are smooth, while billiard flows are not. Although the behaviors of the two systems are close, the theorems often require different proofs in each case: for example, there is no known way of deducing the ergodicity of negatively curved billiards from the ergodicity of geodesic flows on negatively curved surfaces.

A smooth billiard table , where s is the arc length parameter. For example, the walls of a disc are positively curved, while the walls of its complementary set are negatively curved. A billiard whose walls have negative curvature is said to be dispersing.

D in B = T 2 or B = R
One defines the phase space Ω = T1 (Int D), and the billiard flow φ t : Ω → Ω, in the following way:

1. As long as it does not hit a wall, the particle follows a straight line; 2. When it arrives to the boundary of the billiard, the particle bounces, following the billiard reflection law: the angle between the particle's speed vector and the boundary's tangent line is preserved (Figure 6.1).

The flow φ t is not defined at all times :

1. It is not defined at times when the particle is on the boundary of the billiard. Of course, one could extend the definition to such t, but the flow obtained in this way would not be continuous 1 .

2. When the particle makes a grazing collision with a wall at a time t 0 > 0, i.e. collides with the boundary with an angle θ = 0, the flow stops being defined for all times t ≥ t 0 . Although one could extend continuously the definition of the trajectory after such a collision, the differentiability of the flow would be lost. We define Ω as the set of all (x, v) ∈ Ω such that the trajectory starting from (x, v) does not contain any grazing collision, in the past or the future. Notice that Ω is a residual set of full measure, stable under the flow φ t , and that φ t is C ∞ on Ω.

We will say that a billiard has finite horizon if every trajectory hits the boundary at least once.

Uniform hyperbolicity. We are now ready to define the two notions of uniform hyperbolicity which we will use. Definition 6.1. A flow φ : R × M → M on a closed manifold is Anosov (or uniformly hyperbolic) if there exists a decomposition of T M , stable under the flow, (for some a > 0 and λ ∈ (0, 1), which do not depend on x). This definition does not depend on the choice of the Riemannian metric on M .

T x M = E 0 x ⊕ E u x ⊕ E s x where E 0 x = R d dt � � t=0 φ t (x), such that �Dφ t | E s x � ≤ aλ t , �Dφ -t x | E u x � ≤ aλ t
Since the billiard flow is only defined on a non-compact set Ω (dense in Ω), we need another definition for uniform hyperbolicity in the case of billiards. This definition is given in a more abstract framework in [START_REF] Chernov | Chaotic billiards[END_REF], but here we adapt it directly to billiard flows. Definition 6.2. The billiard flow φ t is uniformly hyperbolic if at each point x ∈ Ω, there exists a decomposition of T x Ω, stable under the flow,

T x Ω = E 0 x ⊕ E u x ⊕ E s x where E 0 x = R d dt � � t=0 φ t (x), such that �Dφ t x | E s x � ≤ aλ t , �Dφ -t x | E u
x � ≤ aλ t (for some a > 0 and λ ∈ (0, 1), which do not depend on x).

Structure of the chapter. In this chapter, we explain how to show the uniform hyperbolicity for surfaces of negative curvature and billiards with negatively curved walls, using the cone criterion introduced by Alekseev [START_REF] Alekseev | Quasirandom dynamical systems[END_REF], and its refinement by Wojtkowski [START_REF] Wojtkowski | Invariant families of cones and Lyapunov exponents[END_REF]. The two proofs use exactly the same ideas. The fundamental tool is the study of Jacobi fields.

All surfaces with negative curvature have an Anosov geodesic flow: according to Arnold and Avez [START_REF] Arnold | Problèmes ergodiques de la mécanique classique[END_REF], the first proof of this fact goes back to 1898 [START_REF] Hadamard | Les surfaces à courbures opposées et leurs lignes géodésique[END_REF]. Later, it was extended to all manifolds with negative sectional curvature (a modern proof is available in [START_REF] Katok | Introduction to the modern theory of dynamical systems[END_REF]). But the negative curvature assumption is not necessary for a geodesic flow to be Anosov. To see if a geodesic flow is Anosov, we will need to examine the solutions of the Ricatti equation

u � (t) = -K(t) -u 2 (t)
where K is the Gaussian curvature of the surface, and to use the following criterion: Theorem 6.3. Let M be a closed surface. Assume that there exists m > 0 such that for all geodesic γ : [0, 1] → M , and all u solution of the Ricatti equation along this geodesic such that u(0) = 0, u is well-defined on [0, 1] and u(1) ≥ m. Then the geodesic flow φ t : T 1 M → T 1 M is Anosov. Theorem 6.3 was mentioned in [START_REF] Donnay | Anosov geodesic flows for embedded surfaces[END_REF] and [START_REF] Mls Magalhães | Geometry and dynamics of planar linkages[END_REF], but as far as we know, no detailed proof was available.

In this chapter, we prove Theorem 6.3, and then explain how this result applies in the particular case of closed Riemannian surfaces with negative curvature. We will even show the following refinement: Theorem 6.4. Let M be a closed Riemannian surface with nonpositive curvature. Assume that every geodesic in M contains a point where the curvature is negative. Then, the geodesic flow on M is Anosov. Theorem 6.4 may also be obtained directly, without using Theorem 6.3 or the cone criterion, from Proposition 3.10 of [START_REF] Eberlein | When is a geodesic flow of Anosov type? I,II[END_REF]. Hunt and MacKay [START_REF] Hunt | Anosov parameter values for the triple linkage and a physical system with a uniformly chaotic attractor[END_REF] used this result to exhibit the first Anosov linkage (more details will be given in Chapter 8).

For billiards, we will prove the following counterpart of Theorem 6.4:

Theorem 6.5. If D is a smooth dispersing billiard with finite horizon, then the billiard flow is uniformly hyperbolic in Ω.

In Chapter 7, we will apply Theorem 6.3 to give new examples of surfaces whose geodesic flow is Anosov while their curvature is not negative everywhere.

Consequences of uniform hyperbolicity. It is shown in [START_REF] Pugh | Ergodicity of Anosov actions[END_REF] that (smooth) volumepreserving Anosov flows are ergodic: every invariant subset has either zero or full measure. It was shown later (see [START_REF] Dolgopyat | On decay of correlations in Anosov flows[END_REF] and [START_REF] Klingenberg | Riemannian manifolds with geodesic flow of Anosov type[END_REF]) that Anosov geodesic flows are even exponentially mixing.

As for billiard flows, Sinaï proved ergodicity for smooth dispersing billiards with finite horizon in [START_REF] Sinaȋ | Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards[END_REF]. It was shown in [START_REF] Baladi | Exponential decay of correlations for finite horizon Sinai billiard flows[END_REF] that such flows are exponentially mixing.

Corners and cusps.

In the definition of billiards, it is common to allow the walls to be only piecewise smooth. If the tangent lines at the singularity make a nonzero angle, it is called a corner, otherwise it is called a cusp (see Figure 6.4). Corners and cusps give the possibility for compact billiards in R2 to be dispersing.

Dispersing billiards with corners (and with finite horizon) have uniformly hyperbolic, mixing flows.

When cusps are introduced, the study of the flow becomes more complicated: the first proof of ergodicity was published no sooner than 1995 (see [ Řeh95]). Such billiards still have mixing flows, and the mixing rate is greater than polynomial (see [START_REF] Bálint | Decay of correlations and invariance principles for dispersing billiards with cusps, and related planar billiard flows[END_REF]), but it is unknown whether it is exponential 2 .

The cone criterion

Definition 6.6. Consider a Euclidean space E.

A cone 3 in E is a set C such that there exist a decomposition E = F ⊕ G and a real number α ≥ 0 such that Figure 6.4 -On the left, a dispersing billiard with corners in R 2 . On the right, a dispersing billiard with cusps.

C = {(x, y) ∈ F ⊕ G | �x� ≤ α �y�} .
The number arctan α is called the angle of the cone. Two cones C 1 , C 2 are said to be supplementary if they correspond to decompositions

E = F 1 ⊕ G 1 and E = F 2 ⊕ G 2 such that F 1 = G 2 and F 2 = G 1 .
Proposition 6.7. Consider a sequence of linear mappings A k : R n → R n , and a sequence of supplementary cones C k and D k , corresponding to the decomposition R n = R m × R n-m . Assume that there exist a > 0, λ > 1 such that for all k ∈ Z:

1. A k (C k ) ⊆ C k+1 (invariance in the future), 2. �A k-1 • . . . • A k-i (v)� ≥ aλ i �v� for all i ≥ 0 and v ∈ C k-i (expansion in the future), 3. A -1 k (D k+1 ) ⊆ D k (invariance in the past), 4. � � A -1 k • . . . • A -1 k+i-1 (v)
� � ≥ aλ i �v� for all i ≥ 0 and v ∈ D k+i (expansion in the past).

Then

E u k = +∞ � i=0 A k-1 • . . . • A k-i (C k-i )
is an m-dimensional subspace contained in C k , and

E s k = +∞ � i=0 A -1 k • . . . • A -1 k+i-1 (D k+i ) is an (n -m)-dimensional subspace contained in D k . Proof. For all i ≥ 0, A k-1 • . . . • A k-i (C k-i
) is a cone, which contains a vector space V i of dimension m. Thus, the intersection E u k contains a vector space V of dimension m (for example, consider a converging subsequence of orthonormal bases of V i ). Assume that there exists w ∈ E u k \V . Then there exists v ∈ V and t ∈ R such that v +tw ∈ {0}×R n-m (notice also that tw

∈ E u k ). Since A -1 k-i • . . . • A -1 k-1 (tw) and A -1 k-i • . . . • A -1 k-1 (v) lie in E u k-i , Assumption 2 gives us: � � A -1 k-i • . . . • A -1 k-1 (tw) � � ≤ 1 aλ i �tw� → k→+∞ 0, � � A -1 k-i • . . . • A -1 k-1 (v) � � ≤ 1 aλ i �v� → k→+∞ 0,
but at the same time, since v + tw ∈ D k , Assumption 4 gives:

� � A -1 k-i • . . . • A -1 k-1 (v + tw) � � ≥ aλ i �v + tw� → k→+∞ +∞,
which contradicts the triangle inequality.

One obtains the result for E s k in the same way.

Theorem 6.8.

Let A k = � a k b k c k d k � (with k ∈ Z)
A k v ∈ C � .
Then, there exist a > 0 and λ > 1 such that for all k ∈ Z, for all i ≥ 0 and v ∈ C � , Proof. On the basis of Wojtkowski's idea [START_REF] Wojtkowski | Invariant families of cones and Lyapunov exponents[END_REF], instead of proving expansion directly for the Euclidean norm, we consider the function

�A k-1 • . . . • A k-i (v)� ≥ aλ i �v� .
N : C � → R ≥0 � x y � � → √ xy.
Notice that N is equivalent to the Euclidean norm on C � , i.e. there exists

M > 0 such that for all v ∈ C � , 1 M �v� ≤ N (v) ≤ M �v� , because � 2 (x 2 + y 2 ) ≤ xy ≤ 2 � (x 2 + y 2 ) for all � x y � ∈ C � .
We are going to show that for all k ∈ Z and

v ∈ C � , N (A k v) ≥ 1 1-� 2 N (v).
With the equivalence of norms, this will complete the proof.

Let k ∈ Z. We may assume that det(A k ) = 1, by multiplying A k by � 0 1 1 0 � on the left. Moreover, we may assume that all the coefficients of A k are positive, by multiplying A k by -Id.

Notice that the two vectors

A k � 1 0 � = � a k c k � and A k � 0 1 � = � b k d k � are in the cone C � , by continuity of A k . Then for v = � x y � ∈ C � : N (A k v) = (a k x + b k y)(c k x + d k y) ≥ (a k d k -b k c k )xy + 2b k c k xy ≥ (1 + 2b k c k )N (v) But a k d k -b k c k = 1 and a k ≤ 1 � b k , d k ≤ 1 � c k , so that b k c k ≥ 1 1-� 2 -1. Finally, N (A k v) ≥ 1 1-� 2 N (v).

Anosov geodesic flows

Jacobi fields

To show that a geodesic flow is hyperbolic, one has to study how the geodesics move away from (or closer to) each other. Thus, one considers small variations of a given geodesic. Definition 6.9. Let (M, g) be a Riemannian manifold and γ : (a, b) → M a geodesic. Consider a geodesic variation of γ, i.e. a smooth function

f (t, s) : (a, b) × (c, d) → M
such that f (., 0) is the geodesic γ, and for all s ∈ (c, d), f (., s) is a geodesic. The vector field Y = ∂f ∂s along the curve γ(t) is called an infinitesimal variation of γ.

Proposition 6.10. Any infinitesimal variation of γ is a solution of the Jacobi equation:

Ÿ = -R( γ, Y ) γ,
where R is the Riemann tensor. The solutions of the Jacobi equation are called Jacobi fields.

Proof. Let ∇ be the Levi-Civita connection of (M, g). Since On the other hand (still for s = 0),

R( γ, Y ) γ = R � ∂f ∂s , ∂f ∂t � ∂f ∂t = ∇ ∂f ∂s ∇ ∂f ∂t ∂f ∂t -∇ ∂f ∂t ∇ ∂f ∂s ∂f ∂t -∇ [ ∂f ∂s , ∂f ∂t ] ∂f ∂t = -∇ ∂f ∂t ∇ ∂f ∂s ∂f ∂t . Thus, Ÿ = -R( γ, Y ) γ.
Proposition 6.11. Every Jacobi field along a geodesic γ is an infinitesimal variation of γ.

Proof. Here, we copy the proof of [START_REF] Kobayashi | Foundations of differential geometry[END_REF]. Let t 1 ∈ (a, b). For any t 2 close enough to t 1 , any solution of the Jacobi equation is determined by its values at t 1 and t 2 (since it is a second-order linear equation).

Let Y be a solution of the Jacobi equation along γ. For i = 1, 2, let h i (s) (s ∈ (-�, �)) be a curve such that (h i (0), h � i (0)) = (γ(t i ), X(t i )). If necessary, take a smaller �, and choose t 2 even closer to t 1 , so that there exists for each s a unique geodesic f (., s) (t ∈ (a, b)), through h 1 (s) and h 2 (s), of minimal length between h 1 (s) and h 2 (s). Let X be the Jacobi field ∂f ∂s along γ. Since X and Y are two solutions of the Jacobi equation which coincide at t 1 and t 2 , they are equal. Thus, Y is an infinitesimal variation of γ.

From now on, assume that M has dimension 2, that γ is a unit speed geodesic, and that Y is a Jacobi field which is orthogonal to γ. Choose an orientation of the normal bundle of γ in M (which has dimension 1), i.e. a vector e(t) ∈ T 1 γ(t) M orthogonal to γ � (t), so that Y (t) is identified by one real coordinate, noted y(t) = g(Y (t), e(t)).

The quantity ẏ satisfies In other words, ẏ measures the infinitesimal variation of the vector γ with respect to s. Thus, when y and ẏ have the same sign, the Jacobi field is diverging: the geodesics go away from each other. When y and ẏ have opposite signs, the Jacobi field is converging. We will consider the ratio u = ẏ y , when it is well-defined (i.e. y � = 0), to measure the convergence rate. u > 0 u < 0 Proposition 6.12. When it is well-defined, u is a solution of the Ricatti equation:

u(t) = -K(γ(t)) -u 2 (t).
where K is the Gaussian curvature.

Proof. In dimension 2, the Riemann tensor may be written

�R(a, b)c | d� = K • (g(a, c)g(d, b) -g(a, d)g(c, b)).
Thus, in the case of a unit speed geodesic and of an orthogonal Jacobi field, the vector R( γ, Y ) γ is orthogonal to γ, and its coordinate is Ky. The Jacobi equation then becomes: ÿ = -Ky.

Thus,

u = d dt � ẏ y � = ÿy y 2 - ( ẏ) 2 y 2 = K -u 2 .
The solutions of this equation are not always defined for all times: it may happen that u(t) explodes to -∞ in positive time (or to +∞ in negative time). This corresponds to the phenomenon of convergence of the wavefront: up to order 1, all the geodesics of the infinitesimal variation "gather at one point". In most cases, the Jacobi field becomes divergent just after the convergence point (Figure 6.6).

Proof of Theorem 6.3

Let (x, v) ∈ T 1 M . The tangent plane T (x,v) (T 1 M ) is the direct sum of a vertical and a horizontal subspace H (x,v) ⊕ V (x,v) , given by the metric g on M . Each of these two spaces is naturally endowed with a norm, respectively g H and g V : one equips T 1 M with the norm g T = g H + g V (in particular, one decides that H is orthogonal to V ).

Denote by W (x,v) ⊆ T (x,v) (T 1 M ) the plane orthogonal to the direction of the flow φ t , and let (w, w � ) ∈ W (x,v) . There exists Y (t) a Jacobi field such that (Y (0), Ẏ (0)) = (w, w � ): Figure 6.6 -u is not well-defined at the convergence point.

then the vectors Ẏ (0) and γ(0) are orthogonal, and (Y (t), Ẏ (t)) = Dφ t (w, w � ). Moreover, notice that

∂ ∂t g( Ẏ , γ) = g(∇ ∂f ∂t Ẏ , γ) + g( Ẏ , ∇ ∂f ∂t γ) = g(∇ ∂f ∂s ∇ ∂f ∂t γ, γ) + 0 = 0.
Thus, Y (t) and Ẏ (t) remain orthogonal to γ for all t. In particular, the family of planes (W (x,v) ) (where (x, v) varies in T 1 M ) is stable under Dφ t . Let (x, v) ∈ T 1 M , and γ the geodesic such that (γ(0), γ(0)) = (x, v). Choose an orientation of H (γ(t), γ(t)) ∩ W (γ(t), γ(t)) , i.e. a continuous unit vector e 1 (t) in H (γ(t), γ(t)) ∩ W (γ(t), γ(t)) . It induces naturally an orientation of V (γ(t), γ(t)) , given by a continuous unit vector e 2 (t) in V (γ(t), γ(t)) . This orthogonal basis of W (γ(t), γ(t)) allows us to identify it to the Euclidean R 2 .

For k ∈ Z, set

A k = D (γ(k), γ(k)) φ 1 : W (γ(k), γ(k)) → W (γ(k+1), γ(k+1)) .
The A k are linear mappings with determinant ±1, because the flow φ t preserves the Liouville measure.

We are going to show that the sequence A k : R 2 → R 2 (with the identification above) satisfies the assumptions of Theorem 6.8: with the notations of this theorem, one wants to show that there exists � > 0 such that for all vector v with positive coordinates in R 2 , we have A k v ∈ C � , for some �. In other words, we want to show that there exists � > 0 such that for all geodesics γ : [0, 1] → M and all solutions u of the Ricatti equation u(t) = -K(γ(t))u 2 (t) along this geodesic which satisfy u(0) > 0, u is well-defined on [0,[START_REF] Laetitia | is flat[END_REF] and

� ≤ u(1) ≤ � -1 .
Let K max be the maximum absolute value of the curvature of M , and let � = min(1/4, 1/K max , m). We claim that such an � has the desired property.

The inequality � ≤ u(1) results directly from the main assumption of the theorem, and from the fact that � ≤ m. To show that u(1) ≤ � -1 , we assume that u(1) > � -1 and look for a contradiction. First, notice that for all t ∈ [0, 1] such that u(t

) ≥ � -1 , u � (t) ≤ � -1 -u(t) 2 < 0. Therefore, u(t) ≥ � -1 for t ∈ [0, 1]. From u � ≤ � -1 -u 2 , we obtain u � u 2 ≤ 1 �u 2 -1 ≤ - 1 2 1 u(0) - 1 u(1)
≤ -1 2

6.4 Smooth dispersing billiards

Jacobi fields for billiards

Let D be a billiard and γ : (a, b) → M a billiard trajectory (defined for all times t ∈ (a, b), except for the collision times). Consider a smooth function

f (t, s) : (a, b) × (c, d) → M
such that f (., 0) is the trajectory γ, and for all s ∈ (c, d), f (., s) is a billiard trajectory.

The function f (., s) is not defined at times t 1 (s), t 2 (s), . . . , t ps (s), which correspond to the collision times. By analogy with the case of geodesic flows, we shall call4 "Jacobi field" the vector field Y = ∂f ∂s along the curve γ. Inside the billiard, Y satisfies the equation Ÿ = 0, since the curvature is zero. At a collision time, Y undergoes a discontinuity, which we are now going to study.

Consider a unit speed billiard trajectory variation (i.e. f (., s) is a unit speed billiard trajectory for all s). Denote by τ (s) the first collision time of f (., s). This collision occurs on some piece of the boundary Γ: assume that Γ is parametrized by arc length and define r(s) so that Γ(r(s)) is the point where the collision occurs. The angle between the horizontal axis and

∂ ∂t � � t=t ± 0 f (t, s) is written ω ± (s). Define θ = 1 2 (ω + -ω -) and ψ = 1 2 (ω + + ω -).
The angle θ is the angle of incidence, while ψ is the angle between the horizontal axis and the tangent vector of the boundary Γ � (r(s)).

ψ θ ω + ω -
To state and prove the next result, it is convenient to identify R 2 with C and use complex notation. Proposition 6.14. Denoting by Y -the corresponding Jacobi field just before the collision time, and Y + the Jacobi field just after it,

Y + = -e 2iθ Y -.
Proof. Fix t -< τ and t + > τ . For all small enough s, and for t -and t + sufficiently close to τ :

f (t ± , s) = Γ(r(s)) + (t ± -τ (s))e iω ± (s) .
After derivation with respect to s, evaluating at s = 0: s) .

Y (t ± ) = ∂r ∂s e iψ(s) - ∂τ ∂s e iω ± (s) + i(t ± -τ (s)) ∂ω ± ∂s e iω ± (
Let t + and t -tend to τ :

Y ± = ∂r ∂s e iψ - ∂τ ∂s e iω ± .
Finally, compute:

Y + + e 2iθ Y -= ∂r ∂s
e iψ (1 + e 2iθ ) -2 ∂τ ∂s e iω + = 2 cos θ ∂r ∂s e iω + -2 ∂τ ∂r ∂r ∂s e iω + = 0.

In the case of an orthogonal field, Y may be defined by one scalar y (using the usual orientation of R 2 ). Proposition 6.14 shows in particular that an orthogonal Jacobi field remains orthogonal after a collision. We will write again u = ẏ y . In the interior of the billiard, the trajectories follows the geodesics of B = R 2 or T 2 , whose curvature is zero: thus, the Ricatti equation is simply u � (t) = -u 2 (t). However, when a collision occurs, u undergoes a discontinuity. Proposition 6.15. Consider f a unit speed geodesic variation corresponding to an orthogonal Jacobi field.

At a collision,

y + = -y - ẏ+ = -ẏ-+ 2κ sin θ u + = u -- 2κ sin θ
where κ is the curvature of the boundary and θ is the angle of incidence.

Proof. The equality y + = -y -is a reformulation of Proposition 6.14 in the case of orthogonal Jacobi fields.

On the other hand,

ẏ+ + ẏ-= ∂(ω + + ω -) ∂s = 2 ∂ψ ∂s = 2 ∂ψ ∂r ∂r ∂s = 2κ y - sin θ and u + -u -= ẏ+ y + - ẏ- y -= - ẏ+ + ẏ- y - = - 2κ sin θ .
In particular, positively curved walls decrease the value of u (and tend to make the Jacobi field converge), just as the positive curvature of a Riemannian manifold. Likewise, negatively curved walls have the same effect on u as the negative curvature of a manifold. In particular, notice the following: Proposition 6.16. If D is a dispersing billiard, any orthogonal Jacobi field with u(0) ≥ 0 satisfies u(t) ≥ 0 for all t ≥ 0.

Proof. Assume that there is no collision between t = 0 and some time t = t 1 . Then u is a solution of the equation u

� (t) = -u 2 (t). If u(0) = 0, then u(t) = 0 for all t ∈ [0, t 1 ]. If u(0) > 0, then u � u 2 = -1 for all t ∈ [0, t 1 ] such that u(t) � = 0, so 1 u(0) - 1 u(t) = -t,
which shows that the solution is well-defined and positive for t ∈ [0, t 1 ]. Moreover, Proposition 6.15 shows that u increases at each collision. Therefore, u remains positive at all times.

Proof of Theorem 6.5

The ideas of the proof are the same as for Theorem 6.3. We start with the lemma: Lemma 6.17. Let D be a billiard with finite horizon. Then, there exists t 0 such that every trajectory in D (with unit speed) experiences at least one collision between t = 0 and t = t 0 .

Proof. Assume that the conclusion is false. Then for all n > 0, there exists a billiard trajectory without collision γ n , defined on [-n, n]. By the Arzelà-Ascoli theorem, one may extract a subsequence which converges uniformly on each interval [-n, n]. The limit is still a trajectory without collision, defined on R, which contradicts the assumption.

Notice that for all (x, v) ∈ Ω, T (x,v) Ω is naturally identified with R 2 × R, which is endowed with the standard Euclidean norm. Now, fix (x, v) ∈ Ω and denote by W (x,v) ⊆ T (x,v) (Ω) the orthogonal plane to the direction of the flow φ t . Let (w, w � ) ∈ W (x,v) . There exists an orthogonal Jacobi field Y (t) such that (Y (0), Ẏ (0)) = (w, w � ), and the family of planes W is stable under Dφ t . Let γ be the geodesic such that (γ(0), γ(0)) = (x, v). All planes W (γ(t), γ(t)) , are identified to the same Euclidean R 2 , in the same way as in the proof of Theorem 6.3.

Denote by (t k ) k∈Z the sequence of all collision times of the geodesic γ, and tk = (t k + t k+1 )/2. For k ∈ Z, set:

A k = D (γ( tk ), γ( tk )) φ tk+1 -tk : W (γ( tk ), γ( tk )) → W (γ( tk+1 ), γ( tk+1 )) .
The A k are linear mappings with determinant ±1, because the flow φ t preserves the Liouville measure on Ω (it is possible to check this from Proposition 6.15).

On the interval ]t k , tk [, u is a solution of the equation u � (t) = -u 2 (t), so:

u( tk+1 ) = 1 1 u(t + k+1 ) + ( tk+1 -t k+1 )
. With Propositions 6.15 and 6.16, u(t + k+1 ) ≥ -κ max , where κ max (< 0) is the minimum of the curvature of the walls. Moreover, on a smooth billiard, there exist constants δ min and δ max , independant of k, such that δ min ≤ t k+1t k ≤ δ max for all k ∈ Z. In particular, δ min /2 ≤ tk+1t k+1 ≤ δ max /2, whence:

0 < 1 -1/κ max + δ max /2 ≤ u( tk+1 ) ≤ 1 δ min /2 < +∞.
Thus, Theorem 6.8 applies to the sequence (A k ), and Proposition 6.7 gives the desired result: Theorem 6.5 is proved.

Chapter 7

Geodesic flows of flattened surfaces

Introduction

In 1927, Birkhoff [START_REF] George | Dynamical systems[END_REF] noticed the following fact: if one of the principal axes of an ellipsoid tends to zero, then the geodesic flow of this ellipsoid tends, at least heuristically, to the billiard flow of the limiting ellipse. In 1963, Arnold [START_REF] Igorevich Arnol'd | Small denominators and problems of stability of motion in classical and celestial mechanics[END_REF] stated that the billiard flow in a torus with strictly convex obstacles could be approximated by the geodesic flow of a flattened surface of negative curvature, which would consist of two copies of the billiard glued together, and suggested that this might imply that such a billiard would be chaotic. Later, Sinaï [START_REF] Sinaȋ | Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards[END_REF] proved the hyperbolicity of the billiard flow in this case, without using the approximation by geodesic flows. In the general case, the correspondance between billiards and geodesic flows of shrinked surfaces is well-known, but it is difficult to use in practice. One of the difficulties is the following: near tangential trajectories, some geodesics converge to "fake" billiard trajectories, which follow the boundary of the obstacle for some time and then leave (see Figure 7.1).

More precisely, for a given billiard1 D ⊆ T 2 or D ⊆ R 2 , Birkhoff and Arnold's idea is to consider a surface Σ in a space E = T 2 × R or E = R 3 , such that D = π(Σ), where π Figure 7.1 -A "fake" billiard trajectory, which is the limit of a sequence of geodesics in the flattening surface. The billiard (in grey) is in T 2 and there is one circular obstacle (in white).

is the projection onto the first two coordinates; and then, to consider its image Σ � by a flattening map for � > 0:

f � : E → E (x, y, z) � → (x, y, �z).
The Euclidean metric of R3 induces a metric h � on Σ � . It is convenient to consider the metric g � = (f � ) * (h � ) on Σ, which tends to a degenerate 2-form g 0 on Σ as � decreases to 0. Thus, (Σ, g 0 ) is not a Riemannian metric, but in many cases (for example, for the ellipsoid), it remains a metric space2 and every billiard trajectory in D corresponds to a geodesic in (Σ, g 0 ). The Arzelà-Ascoli theorem guarantees that every sequence of unit speed geodesics (Γ n ) in (Σ, g �n ), with � n → 0, converges to a geodesic of (Σ, g 0 ), up to a subsequence. In this chapter, we prove a stronger version of this result.

Thus, from any given billard, Arnold constructs a surface which he flattens, so that its geodesic flow converges to the billiard flow. In this chapter, we do the reverse: we prove that, under some natural hypotheses, the geodesic flow of any given compact surface in R 3 , or T 2 × R, flattening to a smooth billiard, converges locally uniformly to the billiard flow, away from grazing trajectories (Theorem 7.1). We also prove that, if the limiting billiard has finite horizon and is dispersive, then the geodesic flow in (Σ, g � ) is Anosov for any small enough � > 0 (Theorem 7.2). In this case, it is well-known that the limiting billiard is chaotic, but the surface near the limit does not necessarily have negative curvature everywhere: some small positive curvature may remain in the area corresponding to the interior of the billiard, while the negative curvature concentrates in the area near the boundary. Since the limiting billiard has finite horizon, any geodesic falls eventually in the area of negative curvature, which guarantees that the flow is Anosov. The precise statements of our results are given in Section 7.2.

Other analogies have been made between billiards and smooth dynamical systems. In [START_REF] Turaev | Elliptic islands appearing in near-ergodic flows[END_REF], Turaev and Rom-Kedar showed that the billiard flow could be approximated in the C r topology by the behavior of a particle in R 2 exposed to a potential field which explodes near the boundary 3 . In our situation, there is no potential and the particle has 3 coordinates instead of 2, but some of our techniques are similar to theirs.

Our setting has also much in common with the example of Donnay and Pugh [START_REF] Donnay | Anosov geodesic flows for embedded surfaces[END_REF], who exhibited in 2003 an embedded surface in R 3 which has an Anosov flow. This surface consists of two big concentric spheres of very close radii, glued together by many tubes of negative curvature in a finite horizon pattern. In this surface, any geodesic eventually enters a tube and experiences negative curvature, while the positive curvature is small (because the spheres are big). However, in our situation, we may not choose the shape of the tubes and we need precise estimates on the curvature to show that the geodesic flow is Anosov.

Main results

Let B = R 2 or B = T 2 , and consider a compact surface Σ immersed in E = B × R, whose canonical basis is written (e x , e y , e z ). With the notations of the introduction, we denote by Φ � t the geodesic flow on (Σ, g � ) and by Ψ t the billiard flow on D = π(Σ).

In this chapter, we always assume that D is a smooth billiard: we do not allow corners, to avoid the problems discussed in the introduction. For the definition of billiards, see the beginning of Chapter 6.

Recall that π : E → B is the projection onto the first two coordinates, while f � is a contraction along the z-axis. If Σ is transverse to the fibers of π above Int D, then π and f � induce mappings on the unit tangent bundles:

     π * : T 1 � Σ � ∩ π -1 (Int D) � → T 1 D (q, p) � → � π(q), D q π(p) �D q π(p)� � and      (f � ) * : T 1 Σ → T 1 Σ � (q, p) � → � f � (q), D q f � (p) �D q f � (p)� � .
Consider also the set A of all (t, q, p) ∈ R × T 1 Σ such that π * (q, p) and Ψ t • π * (q, p) belong to Int D, and that the billiard trajectory between π * (q, p) and Ψ t • π * (q, p) does not have a tangential collision with a wall of the billiard. Notice that A is an open dense subset of R × T 1 Σ.

Theorem 7.1. Assume that 1. the surface π -1 (Int D) ∩ Σ is transverse to the fibers of the projection π;

2. for all q ∈ π -1 (∂D) ∩ Σ, the curvature of Σ ∩ V is nonzero at q, where V is a neighborhood of q in the affine plane q + Vect(e z , (T q Σ) ⊥ ).

Then:

A → T 1 (Int D) (t, q, p) � → π * • Φ � t (q, p) converges uniformly on every compact subset of A to A → T 1 (Int D) (t, q, p) � → Ψ t • π * (q, p) as � → 0.
Remark. If Σ is a connected compact surface embedded in R 3 , with positive curvature everywhere, then the two assumptions of Theorem 7.1 are automatically satisfied, and the description of A is simpler.

On the other side, concerning dispersing billiards, we prove: Theorem 7.2. In addition to the two hypotheses of Theorem 7.1, assume that B = T 2 and:

3. the walls of the billiard D have negative curvature; 4. the billiard D has finite horizon: it contains no geodesic of T 2 with infinite lifetime in the past and the future.

Then for any small enough � > 0, the geodesic flow on (Σ � , h � ) is Anosov.

In the proofs of Theorems 7.1 and 7.2, we will assume that Σ is embedded in E, to simplify the notations, but the same proof works for the immersed case. In the case B = T 2 , we will see D as a periodic billiard in the universal cover R 2 , and Σ as a periodic surface in R 3 .

Structure of the proofs. The main tool to study the geodesic flow is the geodesic equation which involves the position q, the speed p, and the normal vector N to Σ:

ṗ = -N (q) �DN (q) • p | p� . (7.1)
It is simply obtained by taking the derivative of the equation

�p | N � = 0.
Equation 7.1 involves the second fundamental form, which is closely linked to the curvature of Σ: in Section 7.3, we make precise estimates on the second fundamental form, study nongrazing collisions with the walls of the billiard (Lemma 7.11) and prove the uniform convergence of the flow (Theorem 7.1). In Section 7.4, we prove that the geodesic flow is Anosov (Theorem 7.2): for this, we also need to study grazing trajectories (Lemma 7.13), and examine the solutions of the Ricatti equation (see Theorem 6.3).

Some questions.

For technical reasons, we had to introduce Assumption 2 in the statements of Theorems 7.1 and 7.2, but is it necessary? One may also wonder whether these theorems generalize to surfaces immersed in higher-dimensional spaces, or if the convergence in Theorem 7.1 holds for the C k topology.

Proof of Theorem 7.1

In Σ ∩ π -1 (Int D), g � converges smoothly to a flat metric, so the geodesic flow converges smoothly to the billiard flow. Hence, the difficulty of the proof concentrates at the boundary of the billiard table: there, we have to show that the geodesic flow satisfies a billiard reflection law at the limit (Proposition 7.11). For this, we will need some estimates on the second fundamental form of the surface near the boundary. First, let us fix some notations.

Definition 7.3. Given � > 0, we may choose a normal vector N � on any simply connected subset of Σ � . We will always assume implicitly that such a choice of orientation has been made: since we work locally on the surface, it is not necessary to have a global orientation.

Consider N � x , N � y , N � z the three components of N � in R 3 . Thus for q ∈ Σ:

N � x (f � (q)) = N 1 x � (N 1 x ) 2 + (N 1 y ) 2 + 1 � 2 (N 1 z ) 2 , N � y (f � (q)) = N 1 y � (N 1 x ) 2 + (N 1 y ) 2 + 1 � 2 (N 1 z ) 2
and

N � z (f � (q)) = 1 � N 1 z � (N 1 x ) 2 + (N 1 y ) 2 + 1 � 2 (N 1 z ) 2 .
We shall often simply write N instead of N � , when there is no possible confusion. Finally, define H(f � (q)) = N 1 z (q). The quantity H(f � (q)) has the advantage of being independent of �, contrary to N � (f � (q)).

For all q ∈ Σ � , we know that π(q) ∈ ∂D if and only if N z (q) = 0, or equivalently, H(q) = 0. This gives us two notions of "being close to the boundary": for all �, δ, ν ∈ (0, 1), we define

V � ν = {q ∈ Σ � | |N � z (f � (q))| < 1 -ν} and Z � δ := {q ∈ Σ � | |H(q)| ≤ δ} .
To simplify the notations, we will often omit the � and simply write V ν and Z δ . Notice that for any δ and ν, when � is sufficiently small, we have V ν ⊆ Z δ , because the metric tends to a flat one outside Z δ .

We now have to deal with three very small values: δ, ν and �. In the whole chapter, we will assume � � δ: in other words, we will first choose a small δ, and then choose a small � which depends on the choice of δ. When the value ν will come into play, its choice will always be made between the choice of δ and the choice of �, that is: � � ν � δ. Definition 7.4 (Darboux frame). For any unit speed curve Γ : [0, 1] → Σ, we define the tangent vector T = Γ � (s). The normal vector N is the unit normal to (T Γ(s) Σ). Finally, the normal geodesic vector G is defined by G = N ∧ T .

In this frame, there exist three quantities γ � N (normal curvature), γ � G (geodesic curvature) and τ � G (geodesic torsion), also written simply γ N , γ G and τ G , such that

dT ds = γ G G + γ N N dG ds = γ G T + τ G N dN ds = γ N T + τ G G The (traditional) curvature of Γ considered as a curve in R 3 is k = � � dT ds � � . Thus, if
k � = 0, writing n = dT /ds �dT /ds� , we obtain:

γ N = k �N | n� , (7.2)
and in particular:

|γ N | = k � 1 -�N | T ∧ n� 2 . (7.3)
For example, if Γ is the intersection of Σ with a plane P , T ∧ n has the same direction as the normal vector of P , so it is convenient to use Equation 7.3. Notice that the normal curvature at s only depends on Γ(s) and Γ � (s): thus we may write γ N (q, p) for (q, p) ∈ T 1 Σ. Moreover, we have the relation:

γ N (q, p) = �DN (q) • p | p� .
For any q ∈ Σ � , γ � + (q) and γ � -(q) (sometimes written simply γ + and γ -) are the principal curvatures of Σ � at q. They correspond respectively to the maximum and minimum normal curvatures at f � (q). K � (q) = γ � + (q)γ � -(q) is the Gaussian curvature of Σ � at q. We can now make a first remark: Fact 7.5. For any small enough δ > 0, H| Z δ is a submersion from Z δ to R.

Proof. Let q ∈ π -1 (∂D) ∩ Σ and consider a curve γ which parametrizes the section of Σ by the plane containing the directions (Oy) and (Oz), with γ(0) = q. Assumption 2 of the theorem implies that γ has nonzero curvature at q, and thus

� D γ(t) H(γ � (t))   γ � (t) �
is nonzero for any small enough t. Therefore, H is a submersion from Z δ to R.

Lemma 7.6. Let (q, p) ∈ T 1 Σ and (q � , p � ) = (f � ) * (q, p). If γ � N (q � , p � ) � = 0 for some � > 0, then the sign of γ � N (q � , p � ) is the same for all � > 0.

Proof. Let Γ : [-1, 1] → Σ be any curve such that (Γ(0), Γ � (0)) = (q, p), and consider

Γ � = f � • Γ for � > 0.
Writing T � its tangent vector, the assumption implies that

� dT � dt   N � � is nonzero at t = 0 for some �, which means that dT � dt � ∈ T Γ � (0) Σ � .
Obviously, this property does not depend on �, so γ � N (q � , p � ) is nonzero for all �. By continuity, γ � N (q � , p � ) does not change sign.

Lemma 7.7. Let α ∈ (0, 1), and q 0 ∈ Σ such that π(q 0 ) ∈ ∂D. We assume that N (q 0 ) is directed towards the exterior of the billiard table D, and (up to a rotation of axis e z ) that N (q 0 ) = -e y . Then there exists r > 0 and � 0 such that for all � ≤ � 0 , all q ∈ B(f � (q 0 ), r), and all p ∈ T 1 q Σ � : γ N (q, p) > 0 whenever |p x | ≤ α (where p x is the x-coordinate of p).

Proof. By Assumption 2 of Theorem 7.1, we know that γ N (q 0 , e z ) > 0. We define

W η = � p ∈ T 1 q 0 Σ   |p x | ≤ η �
. By continuity of γ N , there exists η > 0 such that, for all p ∈ W η , γ N (q 0 , p) > 0.

Write W � η = (f � ) * (W η ). Notice that for all p ∈ T 1 q 0 Σ, writing (f � ) * (q 0 , p) = (f � (q 0 ), p � ):

p � x = p x � p 2 x + � 2 p 2 z and p � z = �p z � p 2 x + � 2 p 2 z .
Thus for a small enough �, W � η contains all p ∈ T 1 q 0 Σ � for which |p x | ≤ α. We denote such an � by � 0 . By Lemma 7.6, γ N > 0 on W � 0 η . Again by continuity, the property γ N > 0 extends to a small neighborhood of the form

� (q, p) ∈ T 1 Σ � 0   q ∈ B(q 0 , r � ), |p x | ≤ α � for some r � > 0.
Finally, we use Lemma 7.6 once again, which proves that there exists r > 0 such that for all � ∈ (0, � 0 ), γ N > 0 on

� (q, p) ∈ T 1 Σ �   q ∈ B(q 0 , r), |p x | ≤ α � .
Proposition 7.8. Choose q 0 ∈ Σ such that π(q 0 ) ∈ ∂D, and assume that N (q 0 ) is directed towards the exterior of the billiard table D, and (up to a rotation of axis e z ) that N (q 0 ) = -e y . Write q � 0 = f � (q 0 ). Then for all α ∈ (0, 1), there exists r 0 > 0 such that for all r ≤ r 0 and for all ν ∈ (0, 1): inf q∈Vν ∩B(q � 0 ,r) p∈T 1 q Σ�, |px|≤α

γ � N (q, p) → �→0 +∞. (7.4)
Moreover, under the additional assumption that the curvature of ∂D is negative at π(q 0 ), there exists r 0 > 0 such that for all r ≤ r 0 and for all ν ∈ (0, 1):

lim sup �→0 sup q∈Vν ∩B(q � 0 ,r) γ � -(q) < 0. (7.5)
To prove this proposition, we first prove a 2-dimensional version in a particular case:

Lemma 7.9. For all � > 0 consider the ellipse

E � = � (y, z) ∈ R 2     y 2 + z 2 � 2 = 1 � .
Define N � (q) as the unit normal vector of the ellipse at q ∈ E � , pointing towards the interior, and let

W ν = {z ∈ E � | |N � z | < 1 -ν} .
Then for all ν ∈ (0, 1), if K(q) denotes the curvature of E � at q:

inf q∈Wν K(q) → �→0 +∞.
Proof. We parametrize E � by:

t � → � cos t � sin t � .
Then the curvature is

K(t) = � (� 2 cos 2 t + sin 2 t) 3/2 while the unit normal vector is � -� cos t -sin t � 1 � � 2 cos 2 t + sin 2 t . If |N z | < 1 -ν, then sin 2 t ≥ (1 -ν) 2 (� 2 cos 2 t + sin 2 t), whence tan 2 t ≤ � 2 (1-ν) 2 ν(2-ν) . Therefore, K = �/ � � cos 3 t � � (� 2 + tan 2 t) 3/2 ≥ � � � 2 + � 2 (1-ν) 2 ν(2-ν) � 3/2 = (ν(2 -ν)) 3/2 � 2
which tends to +∞ as � → 0.

Proof of Proposition 7.8. For each q ∈ B(q 0 , r), consider the curve Γ q resulting from the intersection of Σ with the affine plane (q, e y , e z ) and the normal vector n of Γ q . With the notations of Definition 7.4, let us show that we may choose r small enough for |�N | T ∧ n�| to remain bounded away from 1 for all small � and all q ∈ B(q 0 , r). Since N 1

x (q 0 ) = 0, we may choose r such that N x remains close to 0 for � = 1. We know that N x decreases as � decreases to 0, so N x remains close to 0 when � → 0. Since T ∧ n is colinear to e x , this implies that |�N | T ∧ n�| remains close to 0. Now, let C be a circle tangent up to order 2 to Σ 1 at q, parallel to e ⊥ x : the existence of such a circle is guaranteed, for a small enough r, by Assumption 2 of the theorem. This circle gives birth to a family E � = f � (C) of ellipses which are tangent to Σ � at f � (q) up to order 2. Lemma 7.9 tells us that as � decreases to 0, the curvature of E � at f � (q) (which is the same as the curvature of f � (Γ q ) at f � (q)) tends to infinity as long as q ∈ V ν , uniformly with respect to q. Together with Equation 7.3, this proves that inf q∈Vν ∩B(q � 0 ,r) p∈T 1 q Σ�, px=0

γ � N (q, p) → �→0 +∞.

To prove (7.4), let α ∈ (0, 1). Lemma 7.7 applied to q 0 and α+1 2 gives us some r 0 and � 0 such that for all q ∈ B(q 0 , r 0 ) and all p ∈ T 1 q Σ � such that |p x | ≤ α+1 2 , γ N (q, p) > 0. Since γ N (q, •) is a quadratic form on the tangent space T q Σ � , which takes uniformly large values for p ∈ T 1 q Σ � ∩ (e x ) ⊥ , we deduce that it also takes uniformly large values for

|p x | ≤ α.
Finally, we prove (7.5): consider q ∈ B(q � 0 , r) ∩ V ν , and Γ a parametrization by arc length of {q � ∈ B(q 0 , r) ∩ V ν | H(q � ) = H(q)}. Since H| Z δ is a submersion (for any small enough δ), the curvature of the curve π • Γ is close to the curvature of ∂D near π(q), which is bounded away from zero. Moreover, the unit tangent vector of Γ is bounded away from e z because of Assumption 2, so the speed of π • Γ is bounded away from zero, which implies that the curvature of Γ itself is bounded away from 0, uniformly with respect to � and q. Moreover, �e z | T ∧ n� tends uniformly to 1 as � tends to 0, so �N | T ∧ n� is bounded away from 1 in V � ν . With Equation 7.3, this completes the proof of (7.5).

As a direct consequence of Lemma 7.7 and Proposition 7.8, we obtain: Fact 7.10. If the walls of D are negatively curved, then for any small enough δ, there exists � 0 > 0 such that for � ≤ � 0 , the Gaussian curvature of Σ � in Z δ is negative.

In the following proposition, which is crucial for both Theorems 7.1 and 7.2, we examine nongrazing collisions with the walls of the billiards. We will consider geodesics (q � (t), p � (t)) t∈R in Σ � for some � > 0. We will say that such a geodesic is t 0 -admissible (for some time t 0 > 0) if the billiard trajectory starting from π * (q � (0), p � (0)) bounces against a wall of the billiard before the time t 0 : in this case, we will denote by t c ∈ [0, t 0 ] the time of the first bounce. For any time t 0 > 0, and any t 0 -admissible geodesic, we will denote by (q 0 (t), p 0 (t)) the (unique) pullback of this billiard trajectory by π in Σ � for t ∈ [0, t c ], and write q c = q 0 (t c ); moreover, we will choose the normal N (q c ) so that it is directed towards the exterior of the billiard table D, and make a rotation of axis e z so that N (q c ) = -e y (see Figure 7.2).

Notice that for any δ > 0, the projection of the geodesic flow outside Z δ is uniformly close to the billiard flow for small values of �, so there exists � 0 > 0, which depends on δ, but not on the choice of the geodesic, such that for all � ≤ � 0 , any geodesic as above enters Z δ at a time t � in close to t c .

Proposition 7.11. For all t 0 > 0, m > 0 and α ∈ (0, 1), there exist l < 0 and δ 0 > 0, such that for all δ ≤ δ 0 , there exists � 0 > 0, such that for all � ≤ � 0 and each t 0 -admissible geodesic (q � , p � ) in Σ � such that |p � x (0)| < α:

1. sup t∈[0,t � in + √ δ] |p � x (t) -p � x (0)| ≤ m;
2. the geodesic (q � (t), p � (t)) exits Z δ at a time t out ≤ t in + √ δ;

3. if the curvature of ∂D is negative everywhere, and if q � (0) � ∈ Z δ , then Proof. Fix t 0 > 0, m > 0 and α ∈ (0, 1). In the whole proof, δ 0 and � 0 are assumed to be as small as necessary. In other words, the reader may add mentally to most mathematical sentences the following prefix: "there exist l < 0 and δ 0 > 0, such that for all δ ≤ δ 0 , there exists � 0 > 0, such that for all � ≤ � 0 and each t 0 -admissible geodesic (q � , p � ) in Σ � such that |p � x (0)| < α: . . . ". Let us prove Statement (1). We shall often write q, p for q � (t), p � (t) to simplify the notations.

� t � out t � in K(q � (t))dt ≤ l. N (q c ) V ν Z δ q c e x e y
Outside Z δ , the geodesic flow converges uniformly to the billiard flow, so sup

t∈[0,t in ] |p � x (t) -p � x (0)| ≤ m/2.
Thus, we only need to consider sup t∈[t in ,t in

+ √ δ] |p � x (t) -p � x (t in )|. Let t 1 = inf � t ∈ [t in , t in + √ δ]    �DN (q) • p | p� ≤ 0 � (or t 1 = t in + √ δ if this set is
empty), and consider t ∈ [t in , t 1 ] (thus, �DN (q) • p | p� ≥ 0 at time t).

The geodesics (q(t), p(t)) follow the geodesic equation:

ṗ = -N (q) �DN (q) • p | p� ,
which gives us the following estimates:

|p x (t) -p x (t in )| ≤ � t t in | ṗx | ≤ � t t in � � � � N x N y � � � � | ṗy | ≤ � sup B(q(t in ), √ δ) � � � � N x N y � � � � � � t t in | ṗy |
For all sufficiently small δ, the quantity N y is negative in B(q(t in ), √ δ), thus ṗy = -N y (q) �DN (q) • p | p� is nonnegative and:

|p x (t) -p x (t in )| ≤ � sup B(q(t in ), √ δ) � � � � N x N y � � � � � |p y (t) -p y (t in )| .
We know that Nx Ny = N 1

x N 1 y does not depend on �. Moreover, q(t in ) is close to q c and N 1

x N 1 y (q c ) = 0, so the quantity sup B(q(t in ), ], so it remains bounded away from 1, but γ N (q, p) = �DN (q) • p | p� ≤ 0 at t 1 . Thus, there is a contradiction with Lemma 7.7, and Statement (1) is proved. Now, let us prove Statement [START_REF]) ( i.e. U = M and φ is a bijection), the manifold M is isometric to the Euclidean R q for some q ≥ 0. Proof. Let us prove the first statement. Choose any y ∈ M and four vectors a, b, c, d in T y M of unit length for g. The point is that φ preserves R[END_REF]. We introduce the parameter ν and fix the parameters in the following way: first fix a small δ, then a small ν, and finally a small �.

√ δ) � � � Nx Ny � � �
Let us show that the boundaries of Z δ and V ν near q c are nearly parallel to the e x axis. From Fact 7.5, the levels of (H| Z δ ) -1 (a) are smooth curves. Moreover, for a sufficiently small a ∈ [-1, 1], near q c , the y-coordinates of the unit tangent vectors to H -1 (a) remain small, while the x-coordinates are bounded away from zero. In particular, this applies to the boundary of Z δ , but also to the boundary of V ν , which is a level of H, since N � z depends only on H and � (see Definition 7.3).

Outside V ν , |p z | is bounded by � ν(1ν). Since p x is bounded away from 1 and p 2

x + p 2 y + p 2 z = 1, we deduce that p y remains bounded away from zero, uniformly with respect to δ, ν and �, for all sufficiently small ν. In particular, p y does not change sign in Z δ \ V ν , so it is only possible to enter V ν once with p y < 0 and exit once with p y > 0. Thus, the geodesic can enter V ν at most once.

There remains to show that the time spent in each zone is small. For any q ∈ π -1 (Int D) ∩ Σ � , it is natural to define ∂H ∂x as DH q (p), where p is the (unique) speed vector in T q Σ such that D q π(p) = e x . We also define ∂H ∂y in the same way. Outside V ν (therefore outside π -1 (∂D)) we write:

Ḣ = ∂H ∂x p x + ∂H ∂y p y .
Since the levels of the submersion H are nearly parallel to e x , ∂H/∂x ∂H/∂y is close to 0 near q c . Outside V ν , with the fact that p y is bounded away from 0, this proves that Ḣ is bounded away from 0, so the time spent in Z δ \ V ν is O(δ).

In V ν we have ṗy = -N y �DN (q) • p | p� .

Fix δ, ν > 0. Since N z is bounded away from 1, and Nx Ny is close to zero, we deduce that N y is bounded away from zero. Moreover, by Proposition 7.8, �DN (q) • p | p� → �→0 +∞ uniformly in V ν , so ṗy → �→0 +∞. Since p y is bounded, this implies that the time spent in V ν tends to 0 as � → 0. Thus, the total time spent in each zone is O(δ), so for any small enough δ, t out ≤ t in + √ δ (Statement 2). If the curvature of ∂D is negative everywhere, and q 0 (0) � ∈ Z δ , then the geodesic has the following behavior: it enters Z δ with p y < 0, then enters V ν with p y < 0. In V ν , p y changes sign, then the geodesic exits V ν and finally, exits Z δ . Therefore, writing t 2 and t 3 the entry and exit times in V ν , since K is negative in Z δ (see Fact 7.10):

� t 3 t 2 K = � t 3 t 2 γ � + γ � -≤ � sup Vν γ � - � � t 3 t 2 γ � + ≤ � sup Vν γ � - � � t 3 t 2 -N y �DN (q) • p | p� = � sup Vν γ � - � (p y (t 3 ) -p y (t 2 )).
With Proposition 7.8, this proves that

� t 3 t 2 K is bounded away from 0: Statement (3) is proved.
End of the proof of Theorem 7.1. To prove the local uniform convergence, we introduce a family of elements (t � , q � (0), p � (0)) ∈ f � (A) with parameter �, and assume that � t � , (f � ) -1 * (q � (0), p � (0)) � has a limit (t 0 , q 0 (0), p 0 (0)) ∈ A as � → 0. The geodesic of Σ � starting at (q � (0), p � (0)) is written (q � (t), p � (t)) t∈R . We want to show that π * (q � (t � ), p � (t � )) tends to Ψ t 0 • π * (q 0 (0), p 0 (0)). Since the billiard trajectory Ψ t • π * (q 0 (0), p 0 (0)) experiences only a finite number of bounces in any finite time interval, we may assume that the trajectory for t ∈ [0, t 0 ] has only one bounce 4 , at a time t c . As in Proposition 7.11, let (q 0 (t), p 0 (t)) be the (unique) pullback of this trajectory by π in Σ � for t ∈ [0, t c ], and let q c = q 0 (t c ). Assume that N (q c ) is directed towards the exterior of the billiard table D, and (up to a rotation of axis e z ) that N (q c ) = -e y . The geodesic (q � (t), p � (t)) enters Z δ at some time t � in and exits at some time t � out , and the only difficulty to prove the convergence is located between these two times, since g � converges uniformly to a flat metric outside Z δ .

Since p 0 x (0) < 1, Proposition 7.11 shows that lim

δ→0 lim �→0 |p x (t out ) -p x (t in )| = 0.
Moreover, for all δ > 0, lim �→0 p z (t in ) = lim �→0 p z (t out ) = 0, and since the geodesic has unit speed, lim We have already seen that the geodesic enters Z δ with p y < 0 and exits with p y > 0. Thus: lim

δ→0 lim �→0 p y (t in ) = -lim δ→0 lim �→0 p y (t out ).
Proposition 7.11 also states that lim δ→0 lim �→0 |t outt in | = 0. Thus, the limiting trajectory satisfies the billiard reflection law and Theorem 7.1 is proved.

Proof of Theorem 7.2

In this section, the walls of the billiard are assumed to be concave, and the billiard has finite horizon. The following lemma gives an important consequence of the second property.

Lemma 7.12. Let D be a billiard in T 2 whose walls are negatively curved. Assume that D has finite horizon (D contains no geodesic of T 2 with infinite lifetime in the past and the future). Then, there is an η > 0, a time t max and an angle φ 0 such that every curve of length t max in T 2 , which is η-close to a straight line in the C 1 metric, hits at least once the boundary with an angle ≥ φ 0 .

Proof. Assume that the conclusion of the lemma is false. Then there are curves Γ n : [-n, n] → T 2 which do not hit the boundary with an angle greater than 1 n , and which are 1 n -close to geodesics in the C 1 metric. By a diagonal argument, one may extract a subsequence which converges to a geodesic Γ : R → T 2 which does not hit the boundary with an angle greater than 0, so that Γ remains in D.

As another consequence of the concavity of the walls, we may assume that the principal curvatures satisfy

� � γ � - � � ≤ � � γ � + � � in Z δ (with Proposition 7.8). We write κ(δ, �) = max q� ∈Z δ �� � γ � + (q) � � , � � γ � -(q) � � � .
Notice that for all δ > 0, κ(δ, �) → �→0 0. Later we will simply write κ for κ(δ, �). We also define

W κ = {q ∈ Σ � | K(q) ≤ -κ} .
Notice that for any fixed δ > 0, there exists � 0 > 0 such that for � ≤ � 0 , W κ ⊆ Z δ .

In the following proposition, we determine what remains of Proposition 7.11 when the geodesics are not assumed to be nongrazing, but when instead they are assumed to undergo little curvature. We will consider geodesics (q � (t), p � (t)) t∈R in Σ � , for some � > 0. For such a geodesic, define t � in as the first time at which the geodesic (q � (t), p � (t)) enters Z δ : here, we will will say that a geodesic is admissible if this time exists with t � in + 3 √ δ < 1. As before, we will choose the orientation of the normal vector N such that it points towards the exterior of the billiard table at its boundary near q in , and then make a rotation of axis e z , such that N x (q in ) = 0 and N y (q in ) < 0.

Proposition 7.13. For all m > 0, there exists δ 0 > 0, such that for all δ ≤ δ 0 , there exists � 0 > 0, such that for all � ≤ � 0 and each admissible geodesic (q � , p � ) such that

� 1 0 |K(q � (t))| ≤ 3κ 2 : 1. inf t∈[0,t in + 3 √ δ] (p y (t) -p y (0)) ≥ -2 √ κ; 2. sup t∈[0,t in + 3 √ δ] |p x (t) -p x (0)| ≤ m;
3. denoting by Z 0 δ the connected component of Z δ containing q(t in ), there exists t out ∈ [t in , t in + 3 √ δ], at which the geodesic exits Z 0 δ , and does not come back to Z 0 δ before visiting another component of Z δ . Proof. As before, we only need to consider what happens for t ≥ t in , as the metric tends to a flat metric outside Z δ .

q x (t in ) q x (t out ) O(δ) O( √ δ) Z δ
To prove the first statement, writing ṗy = ṗy +ṗy -, with ṗy + = max( ṗy , 0) (positive part) and ṗy -(t) =min( ṗy , 0) (negative part), it suffices to show that � t t in ṗy -is (uniformly) close to 0. We divide this integral into two parts. In the part where q ∈ W κ , the quantity ṗy = -N y (q) �DN (q

) • p | p� is bigger than - � K(t) (because � � γ � - � � ≤ � � γ � + � � ),
so it is bigger than -K(t) -1. The time spent in W κ is smaller than 3κ 2 /κ, and the integral of |K(t)| is smaller than 3κ 2 . In the part where q � ∈ W κ , �DN (q) • p | p� is bigger than -√ κ. Thus, Finally, to prove Statement 3, fix any α ∈ (0, 1). Lemma 7.11 implies that all trajectories such that |p x (0)| < α exit Z δ before t = t in + √ δ. For the other trajectories, Statement 2 implies that p x remains bounded away from 0. Together with Statement 1 and the uniform concavity of the walls of the billiard table, this implies that the geodesic must exit Z 0 δ definitively before a time which is O( √ δ) (see Figure 7.3).

� t t in ṗy -≤ 3κ 2 + 3κ + √ κ(t in + 3 √ δ) ≤ 2 √ κ,
Lemma 7.14. For all m > 0, there exists � 0 > 0, such that for all � ≤ � 0 , and all geodesics (q � (t), p � (t)) t∈R such that

� 1 0 |K(q � (t))| ≤ 3κ 2 , sup t∈[ 1 3 , 2 3 ] �p(t) -p(1/3)� ≤ m.
In particular, the choice of � 0 does not depend on the choice of the geodesic.

Proof. Outside Z δ , ṗx vanishes as � → 0. Each time that the geodesic enters or exits Z δ , Proposition 7.11 implies (with the choice of α ∈ (0, 1) close to 1) that p is nearly tangent to the boundary of the billiard table (otherwise, the geodesic undergoes strong negative curvature after the entry or before the exit, which is why we consider only the interval

� 1 3 , 2 3

�

). Moreover, Proposition 7.13 implies that the time spent in Z δ is small. Thus, the exit point is near the entrance point and, from Statement 2 of Proposition 7.13, the speed vector p is almost preserved. Then, the geodesic goes to visit another component of Z δ , so there is an upper bound on the number of times that it enters Z δ . Thus, the total change in p is uniformly small.

End of the proof of Theorem 7.2. To show that the flow has the Anosov property, we consider a small δ, a small �, and a geodesic (q � (t), p � (t)) t∈R in Σ � , and examine the Ricatti equation:

� u(0) = 0 u � (t) = -K � (q � (t)) -u 2 (t).
It suffices to show that u(1) is positive and bounded away from 0, uniformly with respect to the choice of the geodesic (see Theorem 6.3). In the following, we write K(t) := K � (q � (t)).

Applying a homothety to Σ if necessary, we may assume that t max given by Lemma 7.12 is less than 1 3 . If

� 1 0 |K(t)| dt ≤ 3κ 2
, then Lemma 7.14 tells us that, for any small enough �, (π * (q(t), p(t))) t∈[ 1 3 , 2 3 ] is C 1 -close to a straight line in T 2 , which contradicts Lemma 7.12. Thus, there exists � 0 > 0 such that for all � ≤ � 0 ,

� 1 0 |K(t)| dt ≥ 3κ 2 . Since K ≤ 0 in Z δ and |K| ≤ κ 2 outside Z δ , we deduce that K ≤ κ 2 in Σ � .
Therefore, considering the positive and negative parts of K,

� 1 0 K = � 1 0 (K + -K -) = - � 1 0 |K| + 2 � 1 0 K + ≤ -3κ 2 + 2κ 2 ≤ -κ 2 .
Now, let us show that u(1) ≥ κ 2 /2, which will end the proof. To do this, we assume that u(1) < κ 2 /2 and show that for all t ∈

[0, 1], |u(t)| ≤ 2κ 2 . Let t 1 = sup � t ∈ [0, 1]   u(t) ≥ 2κ 2 � (or t 1 = 0 if this set is empty). For t ∈ [t 1 , 1], u � (t) = -K(t) -u 2 (t) ≥ -K(t) -4κ 4 , so u(1) -u(t 1 ) ≥ - � 1 t 1 K(t)dt -4κ 4 , whence u(t 1 ) ≤ u(1) + � 1 t 1 K(t)dt + 4κ 4 ≤ κ 2 /2 + κ 2 + 4κ 4 < 2κ 2 .
This implies, with the definition of t 1 , that t 1 = 0 and u(t) ≤ 2κ 2 for all t ∈ [0, 1]. Thus,

u(1) ≥ u(0) - � 1 0 K(t) -4κ 4 ≥ κ 2 -4κ 4 ≥ κ 2 /2,
a contradiction. This ends the proof of Theorem 7.2.

Chapter 8

Dynamics of linkages 8.1 Introduction

In this chapter, we consider the physical behavior of linkages when they are given an initial speed, without any external force and without friction. We will only consider linkages such that Conf(L) is a smooth manifold in (R 2 ) n . It is the case for a generic choice of the edge lengths (see Proposition 1.3). Of course, the dynamics depend on the distribution of the masses in the system: to simplify the problem, we will assume that the masses are all concentrated at the vertices of the graph. If one denotes the speed of each vertex by v i , and the masses by m i , the principle of least action (see [START_REF] Igorevich Arnol'd | Mathematical methods of classical mechanics[END_REF]) states that the trajectory between two times t 0 and t 1 will be a critical point of the kinetic energy

K = 1 2 � t 1 t 0 n � i=1 m i v 2 i (t)dt,
which is also a characterization of the geodesics in the manifold Conf(L) endowed with a suitable metric:

Fact 8.1. The physical behavior of the linkage L, when it is isolated and given an initial speed, is the geodesic flow on Conf(L) ⊆ (R 2 ) n = R 2n , endowed with the metric:

g = n � i=1 m i (dx 2 2i-1 + dx 2 2i ),
provided that the metric g is nondegenerate. In particular, if all the masses are equal to 1, g is the metric induced by the Euclidean R 2n .

Anosov behavior. We ask the following:

Question. Do there exist linkages with Anosov behavior?

The following theorem gives a theoretical answer to this question.

Theorem 8.2. Let (M, g) be any connected compact Riemannian manifold and 1 ≤ k < +∞. Then there exists a linkage L, a choice of masses, and a Riemannian metric h on M , such that h is C k -close to g and every connected component of Conf(L) is isometric to (M, h).

Proof. Embed (M, g) isometrically in some R 2n : this is possible by a famous theorem of Nash [START_REF] Nash | The imbedding problem for Riemannian manifolds[END_REF]. With another theorem of Nash and Tognoli (see [START_REF] Tognoli | Su una congettura di nash[END_REF], and also [START_REF] Ivanov | Approximation of smooth manifolds by real algebraic sets[END_REF], page 6, Theorem 1), this surface is C k -approximated by a smooth algebraic set A in R 2n , which is naturally equipped with the metric induced by R 2n . The manifold A is diffeomorphic to M , and even isometric to (M, h) where the metric h is C k -close to g.

We may now use Theorem 2.2: any compact algebraic set B ⊆ R 2n is exactly the partial configuration space of some linkage, that is, the set of the possible positions of a subset of the vertices; moreover, if B is a smooth submanifold of R 2n , each connected component of the whole configuration space may be required to be smooth and diffeomorphic to B. Thus, there is a linkage and a subset of the vertices W such that the partial configuration space of W is A: each component of the configuration space of this linkage, with masses 1 for the vertices in W and 0 for the others, is isometric to the algebraic set A endowed with the metric induced by R 2n , which is itself isometric to (M, h).

In particular, there exist configuration spaces with negative sectional curvature, and thus with Anosov behavior. This answer is somewhat frustrating, as it is difficult to construct such a linkage with this method in practice, and it would have a high number of vertices anyway, at least several hundreds.

In the 1980's, Thurston and Weeks [START_REF] Willam | The mathematics of three-dimensional manifolds[END_REF] pointed out that the configuration spaces of quite simple linkages could have an interesting topology, by introducing the famous triple linkage (see Section 1.1.3 for more details): they showed that, for some choice of the lengths, its configuration space could be a compact orientable surface of genus 3. Later, Hunt and MacKay [START_REF] Hunt | Anosov parameter values for the triple linkage and a physical system with a uniformly chaotic attractor[END_REF] found out that this particular linkage also had quite interesting dynamics.

Asymptotic configuration spaces. The computation of the curvature of a given configuration space is impossible in practice, most of the time. Thus, the idea of Hunt and MacKay was to give a small length � to some edges1 , while the masses are fixed (0 for the vertex at the center and 1 for the others), and to consider the limit of Conf(L � ) as � → 0. At the limit, the surface is not the configuration space of a physical system anymore (it is called an asymptotic configuration space), but it is easier to study because the equations are simpler. In the case of the triple linkage, the miracle is that the limit surface is Schwarz's well-known "P surface" in T 3 , defined by � 3 i=1 cos x i = 0, which has negative curvature except at a finite number of points, and thus an Anosov geodesic flow (with Theorem 6.4). The structural stability of Anosov flows allows the authors to conclude that the configuration space of L � for a small enough � has an Anosov geodesic flow. In particular, one does not know how small � has to be for L � to be an Anosov linkage.

This technique may be applied to other linkages. For example, in 2013, Pollicott and Magalhães [START_REF] Mls Magalhães | Geometry and dynamics of planar linkages[END_REF] tried to see what happened with the asymptotic "double linkage", an equivalent of the triple linkage but with only two articulated arms (also called "pentagon"). But the asymptotic configuration space in that case has both positive and negative curvature and it is impossible to conclude that the geodesic flow is Anosov, although their computer simulation suggests that it should be the case. In fact, since Hunt and MacKay's example, no other linkage has been proved mathematically to be Anosov.

Linkages and billiards. To understand the link between linkages and billiards, consider Thurston's triple linkage, where all vertices have mass 0 except the central vertex which has mass 1. The workspace of the central vertex is a hexagon (see Figure 1.4), and its trajectories are obviously straight lines in the interior of the workspace, but what happens physically when the vertex hits the boundary of the workspace? It turns out that it reflects by a billiard law. In fact, when the masses of the non-central vertices are small, the configuration space is equipped with the metric of a flattened surface Σ � with a small � > 0, in the spirit of Chapter 7.

However, it may happen that the workspace of some vertex is a dispersive billiard, while the geodesic flow in the configuration space (with a small parameter �) is not Anosov. For example, consider Thurston's triple linkage in the case on the right of Figure 1.4. Then the workspace of the central vertex x is a non-smooth dispersive billiard -a triangle with negatively curved walls -but the configuration space is topologically a sphere, so its geodesic flow cannot be Anosov. In fact, the corners of the billiard concentrate the positive curvature of the configuration space when it flattens.

A new Anosov linkage. In the rest of this chapter, our aim is to give a new example of an Anosov linkage (see Figures 8.1 and 8.2) by applying Theorem 7.2. In this example, the billiard is not the workspace of a single vertex: it is the partial configuration space of four vertices, that is, the set of the possible positions of these vertices. It is a priori a subset of (R 2 ) 4 , but in this particular case, it turns out that it may be seen as a subset of T 2 . The configuration space Conf(L), in turn, may be seen as an immersed surface in T 2 × R which flattens to the billiard table as one of the masses tends to 0 (Figure 8.3). Notice that this example is not an asymptotic linkage, in the following sense: there is a whole explicit range of values for the edge lengths such that the linkage has an Anosov behavior. This is the first time that a linkage with explicit lengths is proved to be Anosov.

However, one mass has to be close to 0 and our theorem does not say explicitly how close it has to be. Maybe this linkage is, in fact, Anosov even when the mass is equal to 1. Notice that the masses in Hunt and MacKay's example are also unexplicit.

In our example, no vertex is fixed, but all the vertices (except one) have only one degree of freedom and move on a straight line. This may be realized physically by prismatic joints -or, if one wants to stick to the traditional definition of linkages, it is possible to use Peaucellier's straight line linkage, or to approximate the straight lines by portions of arcs of large radius.

Precisely, we prove the following:

Theorem 8.3. In the linkage of Figure 8.2, choose the lengths of the rods such that l + r > 3, l < 3, (l -2) 2 + r 2 < 1 and r < 1/2. The mass at (a, 0), (-2, f ), (b, 0) and (2, g) is 1, the mass at (0, c) is � 2 , while the mass at (d, e) is 0. Then for any sufficiently small � > 0, the geodesic flow on the configuration space of the linkage is Anosov. A realistic physical system. Similarly to Hunt and MacKay [START_REF] Hunt | Anosov parameter values for the triple linkage and a physical system with a uniformly chaotic attractor[END_REF], we insist on the fact that our linkage is realistic from a physical point of view. For example, it is possible to add small masses to the rods and to the central vertex without losing the Anosov property (using the structural stability of Anosov flows). See Hunt and MacKay's article for more details about this aspect.

Proof of Theorem 8.3

The aim of this section is to prove that the configuration space of the linkage described in Theorem 8.3 is isometric to an immersed surface in T 2 × R which satisfies the 4 assumptions of Theorem 7.2. The configuration space Conf(L) is the set of all (a, b, c, d, e, f, g) ∈ R 7 such that:

(a + 2) 2 + f 2 = (b -2) 2 + g 2 = 1; (a -d) 2 + e 2 = (b -d) 2 + e 2 = l 2 ; d 2 + (c -e) 2 = r 2 .
Notice that (a + 2, f ) and (b -2, g) lie in the unit circle T ⊆ R 2 . Thus, Conf(L) is in fact a subset of T 2 × R 3 and any of its elements may be written (θ, φ, c, d, e), with the identification a =cos θ -2, f = sin θ, b = cos φ + 2, g = sin φ.

: V → R 3 such that U = {(D, F (D)) | D ∈ V } .
Proof. The function F is given by the following formulae:

d = -cos θ + cos φ 2 ; e = ± � l 2 - � cos θ + cos φ 2 + 2 � 2 = ± � l 2 - � cos θ + cos φ + 4 2 � 2 ; c = e ± � r 2 - � cos θ -cos φ 2 � 2 = ± � l 2 - � cos θ + cos φ + 4 2 � 2 ± � r 2 - � cos θ -cos φ 2 � 2
where the choices of the signs are made according to C 0 . Proof. By symmetry, we only need to prove the first statement. The idea of the proof is the same as for Fact 8.4: on the one hand, the numbers d and e are obtained as the simple roots of a polynomial of degree 2, so they vary smoothly with respect to θ and c; on the other hand, φ = ± arccos(2d + cos θ) where the choice of the sign is made according to C 0 .

Fact 8.6. For all C 0 ∈ Conf(L), Conf(L) is locally a smooth graph near C 0 :

1. either above θ and φ, 2. or above θ and c, 3. or above φ and c.

Proof. Assume the opposite. Then the hypotheses of Fact 8.5 are not satisfied. If φ = θ = 0 mod π, then φ = θ mod 2π because r < 1/2; then φ = θ = π mod 2π because l < 3, but this implies that e � = 0 and e � = c, so Fact 8.4 applies, which is impossible. Therefore, by symmetry, we may assume that (-cos θ -2, 0), (d, e) and (0, c) are aligned. Now, with Fact 8.4, we have either e = 0 or e = c. In both cases, (-cos θ -2, 0), (d, e) and (0, c) are all on the line y = 0, which contradicts the fact that l + r > 3.

Fact 8.6 implies in particular that Conf(L) is a smooth submanifold of T 2 × R 3 . As explained in the introduction of the chapter, Conf(L) is endowed with the metric which corresponds to its kinetic energy (recall that the masses of the vertices are � 2 at (0, c), 0 at (d, e), and 1 everywhere else):

g � = da 2 + df 2 + db 2 + dg 2 + � 2 dc 2 = dθ 2 + dφ 2 + � 2 dc 2 .
Proof. Here we will assume that π

(q) ∈ � (θ, φ) ∈ T 2   cos φ + cos θ = 2l -4 � , but the proof is identical for the other components of ∂D. Let F (θ, φ) = � cos θ+cos φ+4 2 � 2
. For any small t ≥ 0, let θ(t) = q θ + N θ (q)t, φ(t) = q φ + N φ (q)t, and choose z(t) of the form:

z(t) = ± � l 2 - � cos θ(t) + cos φ(t) + 4 2 � 2 ± � r 2 - � cos θ(t) -cos φ(t) 2 � 2
with a choice of the ± signs so that (θ(0), φ(0), z(0)) = q. Then for all small t ≥ 0, (θ(t), φ(t), z(t)) ∈ Σ.

As t tends to 0, we may estimate:

z(t) = ± � � d dt � � � � t=0 F (θ(t), φ(t)) � t + O(t 2 ) ± � r 2 - � cos θ(0) -cos φ(0) 2 � 2 + O(t) (z(t) -z(0)) 2 = ± � d dt � � � � t=0 F (θ(t), φ(t)) � t + o(t). Notice that ∇F (θ(0), φ(0)) = - � sin θ(0) sin φ(0) � � cos θ+cos φ+4 2 � is nonzero (because 2 < l < 3). Moreover, � θ � (0) φ � (0) � is � N θ (q) N φ (q) � , which is colinear to ∇F (θ(0), φ(0)), so d dt � � � � t=0 F (θ(t), φ(t)) = � θ � (0) φ � (0) � • ∇F (θ(0), φ(0)) � = 0.
This gives us:

t = ± 1 d dt � � t=0 F (θ(t), φ(t)) (z(t) -z(0)) 2 + o((z(t) -z(0)) 2 ).
Hence, t � → z(t) has an inverse function z � → t(z) which has a nonzero second derivative at t = 0. Since (z, t) are the coordinates in an affine (orthonormal) basis of q + Vect(e z , (T q Σ) ⊥ ), this implies that Σ ∩ V has nonzero curvature at q.

The following proposition proves Assumption 3.

Proposition 8.8. The walls of the billiard D have negative curvature.

Proof. In general, the curvature of the boundary of a set defined by the inequality F (q) ≤ C, where C ∈ R is a constant, with the normal vector pointing inwards, is the divergence of the normalized gradient of F , namely:

∇ • ∇F �∇F � .
First consider the boundary of the set {cos φ + cos θ ≤ 2l -4}. Here F (φ, θ) = cos φ + cos θ. Thus:

∇F �∇F � = -1 � sin 2 θ + sin 2 φ � sin θ sin φ � .
Hence, the divergence of the normalized gradient has the same sign as:

sin 2 φ cos θsin 2 θ cos φ which can be rewritten:

-(2l -4) cos 2 θ + (2l -4) 2 cos θ -(2l -4).
This is a second order polynomial in cos θ with discriminant (2l -4) 2 ((2l -4) 2 -4) < 0 (here we use the assumption l < 3). Since (2l -4) > 0 (because l > 2), the polynomial is everywhere negative. Now, consider the boundary of the set {cos φcos θ ≤ 2r}. This time, the divergence of the normalized gradient has the same sign as sin 2 φ cos θsin 2 θ cos φ which can be rewritten -2r cos 2 θ -4r 2 cos θ -2r.

This time, the discriminant is 16r 2 (r 2 -1), which is negative since r < 1.

The third wall is the boundary of the set {cos θcos φ ≤ 2r}. The divergence of the normalized gradient has the same sign as sin 2 φ cos θ + sin 2 θ cos φ which can be rewritten -2r cos 2 θ + 4r 2 cos θ -2r.

Again, the discriminant is 16r 2 (r 2 -1), which is negative.

Finally, we prove Assumption 4, which will end the proof of Theorem 8.3.

Proposition 8.9. If (l -2) 2 + r 2 < 1 and r < 1/2, then D has finite horizon.

Proof. Assume that there exists a geodesic (θ(t), φ(t)) with infinite lifetime in the past and in the future. First, we prove that the slope of the geodesic is ±1. We may assume that the slope is in [-1, 1] (if not, exchange the roles θ and φ): thus there is a time t 0 for which θ(t 0 ) = 0 mod 2π. Then the set G

= {φ(t) -φ(t 0 ) mod 2π | t ∈ R, θ(t) = 0 mod 2π} is a subgroup of R/2πZ. Moreover, for all t ∈ G, we have |cos θ(t) -cos φ(t)| ≤ 2r so cos φ(t) ≥ 1 -2r > 0, so G ⊆ � -π 2 , π 2
� mod 2π, which means that G is reduced to a single point: the slope is either 0 or ±1 (since we assumed it is in [-1, 1]). If the slope is 0, then |cos θ(t)cos φ(t)| ≤ 2r applied to a t such that cos θ(t) = -1 gives us cos φ(t) ≤ -1 + 2r, which is not compatible with cos φ(t) ≥ 1 -2r > 0 since r < 1/2, so the slope is in fact ±1.

Changing θ into -θ if necessary, we may assume that the slope is 1. Thus, there exist t 1 and t 2 such that φ(t 1 ) + θ(t 1 ) = π mod 2π and φ(t 2 ) + θ(t 2 ) = 0 mod 2π. We have θ(t 2 )θ(t 1 ) = φ(t 2 )φ(t 1 ) mod 2π (because the slope is 1), so φ(t 2 )φ(t 1 ) = π 2 mod π, so cos φ(t 2 ) cos φ(t 1 ) =sin φ(t 2 ) sin φ(t 1 ). By taking the squares of both sides of this equality we obtain: cos 2 φ(t 1 ) + cos 2 φ(t 2 ) = 1.

(8.1)

We havecos θ(t 1 ) + cos φ(t 1 ) ≤ 2r and cos φ(t 2 ) + cos θ(t 2 ) ≤ 2l -4, which implies that cos θ(t 1 ) = cos φ(t 1 ) ≤ r and cos θ(t 2 ) = cos φ(t 2 ) ≤ l -2. Injecting this in (8.1), we obtain: r 2 + (l -2) 2 ≥ 1, which contradicts r 2 + (l -2) 2 < 1.

Part III

Transverse similarity structures on foliations Theorem 9.1 [START_REF] De | Sur la reductibilité d'un espace de Riemann[END_REF]. Consider a complete connected Riemannian manifold (M, g). Then the universal cover ( M , g) of (M, g) admits the following decomposition:

( M , g) = (M 0 , g 0 ) × (M 1 , g 1 ) × . . . × (M k , g k ),

where k ≥ 0, M 0 is the Euclidean space R q (for some q ≥ 0), and M 1 , . . . , M k are non-flat, locally irreducible manifolds. Moreover, π 1 (M ) acts on M as a subgroup of Isom(M 0 , g 0 ) × Isom(M 1 , g 1 ) × . . . × Isom(M k , g k ).

This decomposition is unique up to the order of the factors.

One of the aims of this chapter is to extend this theorem to the case of a manifold endowed with a locally metric connection, i.e. a connection whose restricted holonomy group Hol 0 (∇) is a relatively compact subgroup of GL n (R). Such a connection is locally the Levi-Civita connection of a Riemannian metric. Moreover, its pullback ∇ to the universal cover M of M has a relatively compact holonomy group, so ∇ is globally the Levi-Civita connection of some Riemannian metric g on M .

Example 9.2. Consider N = R n \ {0} with the Levi-Civita connection ∇ N of the Euclidean metric, and the subgroup G of Diffeo(N ) spanned by the homothety ϕ : x � → 2x. Since ϕ preserves ∇ N , M = N/G is naturally endowed with a connection ∇, which is locally metric, but not globally.

Locally metric connections behave well with respect to the product structure: if (M 1 , ∇ 1 ) and (M 2 , ∇ 2 ) are two manifolds with locally metric connections, then the product connection (∇ 1 , ∇ 2 ) is again a locally metric connection on M 1 × M 2 . In this framework, we will prove the following theorem, which is an analogue of Theorem 9.1: Theorem 9.3. Consider a compact connected manifold (M, ∇), where ∇ is a locally metric connection. Then its universal cover ( M , ∇) admits the following decomposition:

( M , ∇) = (M 0 , ∇ 0 ) × (M 1 , ∇ 1 ) × . . . × (M k , ∇ k )
where k ≥ 0, M 0 is flat, and M 1 , . . . , M k are non-flat, locally irreducible manifolds. Moreover, π 1 (M ) acts on M as a subgroup of

Aff(M 0 , ∇ 0 ) × Aff(M 1 , ∇ 1 ) × . . . × Aff(M k , ∇ k ).
This decomposition is unique up to the order of the factors.

Here, Aff(M, ∇) is the group of affine transformations of M , i.e. diffeomorphisms which preserve the connection ∇.

Notice the two main differences between the statements of Theorems 9.1 and 9.3:

1. In Theorem 9.3, M is assumed to be compact instead of complete. In fact, a manifold with a locally metric connection is almost never geodesically complete.

2. In Theorem 9.3, the flat manifold M 0 is not necessarily a Euclidean space. In Example 9.2, the manifold M is flat, but its universal cover is a covering of the Riemannian manifold R n \ {0} which is not complete, so M 0 cannot be a Euclidean space. By contrast, in the setting of Theorem 9.1, the universal cover is always complete.

We prove Theorem 9.3 in Section 9.4. The difficulty lies in the fact that, although there always exists a global metric on M which is preserved by ∇, it is (most of the time) not complete, so that Theorem 9.1 does not apply. Instead, we will use the following generalization of De Rham's theorem: Theorem 9.4 [START_REF] Ponge | Twisted products in pseudo-Riemannian geometry[END_REF]. Let M be a simply connected Riemannian manifold, whose Levi-Civita connection ∇ is reducible: thus, the tangent bundle T M admits two complementary orthogonal distributions E � and E �� invariant by parallel transport, which induce foliations F � and F �� . Assume that the leaves of F � are all complete. Then, M is globally isometric to a product of Riemannian manifolds M � × M �� . Theorem 9.4 is a consequence of the main result of [START_REF] Ponge | Twisted products in pseudo-Riemannian geometry[END_REF], but we will give a direct proof in Appendix A, based on the ideas of the proof of the classical De Rham theorem in [START_REF] Kobayashi | Foundations of differential geometry[END_REF].

Conformal structures

Next, we focus on a particular case of locally metric connections: those which preserve a conformal structure.

Such a connection appears naturally on the quotient of a Riemannian manifold by a group of similarities, like in Example 9.2. It is essential to notice that these connections do not behave well with the product structure: the product of two such connections does not preserve a conformal structure in general. There is no natural notion of product for conformal structures: thus, it is much more complicated to construct examples of such connections which are reducible.

In this case, the De Rham decomposition takes a very particular form. In fact, it was conjectured recently (see [START_REF] Belgun | On the irreducibility of locally metric connections[END_REF]) that a locally (but not globally) metric connection which preserves a conformal structure was either flat or irreducible. Indeed, it is the case for "Riemannian cones", as proved by Gallot (see Theorem 9.11). Very recently, however, Matveev and Nikolayevsky gave a counterexample to this conjecture [START_REF] Matveev | A counterexample to Belgun-Moroianu conjecture[END_REF], which corresponds to k = 1 and M 0 = R in the setting of Theorem 9.3. Then, they proved the following [MN15b]: Theorem 9.5 (Matveev-Nikolayevsky, 2015). Consider a compact connected manifold (M, ∇), where ∇ is an analytic, locally metric connection which preserves a conformal structure, but is not globally metric.

Then either (M, ∇) is flat, or its universal cover ( M , ∇) admits the following decomposition:

( M , ∇) = (M 0 , ∇ 0 ) × (M 1 , ∇ 1 )

where M 0 = R q (q ≥ 0), and M 1 is a non-flat, locally irreducible manifold. In this case, π 1 (M ) acts on M as a subgroup of Aff(M 0 , ∇ 0 ) × Aff(M 1 , ∇ 1 ).

We answer positively, by a new proof, to a question asked in [START_REF] Vladimir | Locally conformally berwald manifolds and compact quotients of reducible manifolds by homotheties[END_REF]:

Theorem 9.6. Theorem 9.5 remains true without assuming analyticity.

In other words, the De Rham decomposition of a locally metric connection given in Theorem 9.3 takes a particular form when the connection preserves a conformal structure: the number k of irreducible factors is always 0 or 1, and if k = 1, then M 0 = R q for some q ≥ 0.

Similarity structures

Locally metric connections which preserve a conformal structure have an important property: they preserve similarity structures. Definition 9.7.

1. On a differentiable manifold M , a similarity structure is given by an open cover (U i ) 1≤i≤r of M , together with a Riemannian metric g i on each U i , such that the transitions are locally similarities, i.e.

g j = λ ij g i on U i ∩ U j , (9.1)
where each λ ij is locally constant. Of course, any Riemannian structure induces a similarity structure. Notice also that any similarity structure induces a conformal structure in a natural way.

2. If (M, g) is a Riemannian manifold, its similarity pseudogroup Sim loc (M ) consists of all φ : U → V such that φ * g = λg, where U and V are open subsets of M , and λ ∈ R >0 is locally constant on U . For any x ∈ M , the number λ(x) is called the ratio of φ at x (if M is connected, there is no need to specify the point x). The similarity group Sim(M ) is the set of all φ ∈ Sim loc (M ) which are bijections from the whole manifold M to itself.

Here are three fundamental examples:

Example 9.8. Consider a Riemannian manifold (N, g N ), its Levi-Civita connection ∇ N , and G a subgroup of Sim(N ) acting properly discontinuously on N . Then the quotient M = N/G is naturally endowed with a similarity structure.

Example 9.9. Let (M, g) be a Riemannian manifold, and ( M , g) its universal cover. Any closed 1-form ω on M lifts to an exact 1-form ω on M . Consider a primitive f of ω and let h = e f g. Then the fundamental group of M acts on ( M , h) by similarities, and thus h induces a similarity structure on M (see Example 9.8).

Example 9.10. Let N be any Riemannian manifold. The cone over N is the manifold M = N × S 1 endowed with the similarity structure given by Example 9.9, where ω is the canonical form dθ on M .

Levi-Civita connection. Any similarity structure defines canonically a "Levi-Civita" connection on M : on each U i , it is obtained by taking the Levi-Civita connection of g i . Equation 9.1 ensures that this connection is well-defined. From this viewpoint, we may state Gallot's result [START_REF] Gallot | Équations différentielles caractéristiques de la sphère[END_REF], which was mentioned above: Theorem 9.11 [START_REF] Gallot | Équations différentielles caractéristiques de la sphère[END_REF]. If (M, ∇) is the cone over a Riemannian manifold N (see Example 9.10), then ∇ is either irreducible or flat.

Moreover, the Levi-Civita connection of a similarity structure is locally metric and preserves the conformal structure induced by the similarity structure.

Conversely, it turns out that all locally metric connections which preserve a conformal structure are obtained as Levi-Civita connections of similarity structures. More precisely: Fact 9.12. If (M, ∇) is a connected manifold endowed with a locally metric connection which preserves a conformal structure, then (M, ∇) is the quotient of a Riemannian manifold (N, g) (endowed with its Levi-Civita connection) by a subgroup G of Sim(N ). In other words, ∇ is the Levi-Civita connection of a similarity structure on M .

Proof. Consider a conformal structure c on M which is preserved by ∇, a point x ∈ M , and a metric g in the conformal class c defined in a neighborhood of x. Since ∇ preserves c, all the elements of the restricted holonomy group Hol 0

x (∇) are similarities of the Euclidean space (T x M, g x ). Moreover, since ∇ is locally metric, Hol 0

x (∇) is relatively compact, so its elements are in fact isometries. Now, consider the universal cover ( M , ∇) of (M, ∇), x ∈ M an element which projects to the point x, c the pullback of the conformal class c, and g the pullback of the metric g in a neighborhood of x. The elements of Hol x( ∇)(= Hol 0

x( ∇)) are also isometries of the Euclidean space (T x M , gx ). Thus, by parallel transport of gx , one obtains a metric h defined on the whole manifold M , and this metric is in the conformal class c. Let α ∈ π 1 (M ). Since α preserves the conformal class of h, there exists λ : M → R >0 such that α * h = λ h. Moreover, h is invariant under the parallel transport of ∇, and ∇ is preserved by π 1 (M ), so λ is constant. Hence, π 1 (M ) acts by similarities on M .

Irreducible Levi-Civita connections have special properties: Fact 9.13. Consider a Riemannian manifold (M, g) with its Levi-Civita connection ∇. If ∇ has irreducible restricted holonomy, then:

1. the only metrics whose Levi-Civita connection is ∇ are the metrics h λ = λg, λ > 0;

2. Aff(M, g) = Sim(M, g).

Proof.

1. Let h be a metric with whose Levi-Civita connection is ∇ and let x ∈ M . Define the linear mapping F : T x M → T x M in the following way: for all u ∈ T x M , F (u) is the unique vector such that g(u, •) = h(F (u), •). Since Hol x (∇) preserves g and h, all the elements of Hol x (∇) commute with F , so the eigenspaces of F are stable under Hol x (∇). Since ∇ is irreducible, the only possible eigenspaces for F are {0} and T x M ; but F is self-adjoint (for both metrics g and h), so F is a homothety, and thus g and h are proportional.

2. For all φ ∈ Aff(M, g), the metric φ * g is preserved by the Levi-Civita connection ∇ of g, so φ * g is proportional to g and thus φ is a similarity.

Fact 9.13 implies that any irreducible locally metric connection on a manifold M preserves a conformal structure (indeed, any metric g which is preserved by ∇ on the universal cover ( M , ∇) induces a conformal structure on M which is preserved by ∇). Moreover, we now have a more precise version of Theorem 9.3: Theorem 9.14. Consider a compact connected manifold (M, ∇), where ∇ is a locally metric connection. Then its universal cover ( M , ∇) admits a metric g which is preserved by ∇, such that:

( M , g) = (M 0 , g 0 ) × (M 1 , g 1 ) × . . . × (M k , g k ) on this ball is between 1/m and m (because x 0 ∈ E). Therefore, γ is well-defined on f i 1 (x 0 ), and its ratio is between 1/m and m at f i 1 (x 0 ). The ratio of γ is ≥ 1/(k 2 max m) at f i (y 0 ), so y 0 ∈ E, and E is open. Now, we show that M \ E is open in M . Let x 0 ∈ M \ E, i ∈ {1, . . . , r}, and γ ∈ Γ defined on f i (x 0 ) with ratio < 1/m. Then γ is defined on a connected open set W ⊆ T i containing f i (x 0 ), and f -1 i (W ) is an open set of M , containing x 0 and contained in M \ E, so M \ E is open.

End of the proof of Theorem 9.15. Assume that (T, g) is not flat, and let T � be the set of all y ∈ T at which the Riemann tensor of g is nonzero. Notice that T � is stable under the holonomy pseudogroup Γ. Now, Proposition 9.19 gives us a Riemannian metric g � on T � which is invariant by Sim loc (T � ), and thus by the holonomy pseudogroup Γ. Hence, the set E defined in Lemma 9.23 is non-empty. By Lemma 9.23, Γ is equicontinuous. Finally, in view of Proposition 9.18, F is a Riemannian foliation, and Theorem 9.15 is proved.

End of the proofs of the main results

Let us start this section with the proof of Theorem 9.3. Consider a compact manifold (M, ∇), where ∇ is locally metric, and its universal cover ( M , ∇), on which there is a metric g preserved by ∇. Fix x ∈ M and choose a preimage x ∈ M .

Let E 0 x be the maximal linear subspace of the tangent space T x M on which Hol x( ∇) acts trivially. Let E >0

x be the orthogonal complement of E 0 x. The local theorem of De Rham (see for example [START_REF] Kobayashi | Foundations of differential geometry[END_REF]) states that there is a unique decomposition of E >0

x (up to the order of the factors) into mutually orthogonal, invariant irreducible subspaces:

E >0 x = E 1 x ⊕ . . . ⊕ E k x .
This induces a decomposition T x M = E 0

x ⊕ E >0 x , where E >0 x = E 1 x ⊕ . . . ⊕ E k x . Moreover, since π 1 (M ) acts on M by preserving the connection ∇, this decomposition does not depend on the choice of the preimage x of x, up to the order of the factors. Thus, the holonomy group Hol x (∇) acts on E >0

x by permuting the factors: by considering a finite cover of M , one may assume that Hol x (∇) preserves the decomposition of T x M . Then, one may consider E � the distribution on M obtained by parallel transport of E k x , and E �� obtained by parallel transport of E 0

x ⊕ . . . ⊕ E k-1

x . These distributions induce transverse foliations F � and F �� on M .

There exists a covering (U i ) 1≤i≤r of M compatible with the foliations F � and F �� , such that each U i is diffeomorphic to V i × T i , where V i (the plaque of F �� ) is an open ball of R p and T i (the plaque of F � ) an open ball of R q . The connection ∇ induces a connection ∇ T on the transversal T = ∪ 1≤i≤r T i , which is preserved by the holonomy pseudogroup. Since T is simply connected, ∇ T preserves a Riemannian metric g T . The holonomy pseudogroup of F �� acts by affine transformations on (T, ∇ T ). But since ∇ T is irreducible, these transformations are in fact local similarities of (T, g T ), which implies that F �� has a transverse similarity structure.

By construction, the holonomy group of M does not act trivially on E � , so F �� is not transversally flat. With Theorem 9.15, F �� is transversally Riemannian, so the leaves of F � are naturally endowed with a Riemannian structure. In particular, the leaves of the pullback F� of F � are all complete. Therefore, by Theorem 9.4, M is the product of two Riemannian manifolds M � and M �� . The existence of the decomposition given in Theorem 9.3 follows by induction on the dimension of M .

Uniqueness is proved in the same way as for the usual decomposition theorem of De Rham: the factors are necessarily the maximal integral manifolds of the distributions given by E 0

x, E 1 x, . . . , E k x . Thus, Theorem 9.3 is proved. For the proof of Theorem 9.6, we will need the following propositions: Proposition 9.24. Consider a complete connected Riemannian manifold (M, g). If Sim(M ) does not act properly on M , then M is (globally) isometric to R q for some q ≥ 0.

Proof. In this case, there exist a compact set K ⊆ M and a sequence (S n ) of similarities such that K ∩ S n (K) � = ∅ and the ratio of S n (written r n ) tends to +∞ or 0 when n → +∞. Considering S -1 n instead of S n if necessary, we may assume that r n → 0. Let K � = {x ∈ M | d(x, K) ≤ �} for some small � > 0, where d is the distance induced by g in M . Then S n (K � ) = {x ∈ M | d(x, S n (K)) ≤ r n �}: in particular, for some large enough n 0 > 0, S n 0 (K � ) ⊆ K � . Thus, S n 0 has a fixed point and M is isometric to R q by Proposition 9.20. Proposition 9.25. Consider the product of two connected Riemannian manifolds, denoted by (M, h) = (M 1 , h 1 ) × (M 2 , h 2 ), and a subgroup G of Sim(M ) which preserves the product structure ( i.e. which is a subgroup of Sim(M 1 ) × Sim(M 2 )), and acts on M in a cocompact way. Also assume that Sim(M ) contains elements which are not isometries. Then, either M 1 = R q or M 2 = R q , for some q ≥ 0.

Proof. Assume that the conclusion is false. In view of Lemma 9.24, Sim(M 1 ) and Sim(M 2 ) act properly on M 1 and M 2 respectively.

Since G acts cocompactly on M , there is a compact set K ⊆ M such that Sim(M )•K = M . We may assume that K = K 1 ×K 2 , where Sim(M 1 )•K 1 = M 1 and Sim(M 2 )•K 2 = M 2 .

Choose x 1 ∈ K 1 . Since Sim(M 1 ) acts properly on M 1 , there is a constant R > 1 such that for all γ ∈ Sim(M 1 ) satisfying γ(x 1 ) ∈ K 1 , the ratio of γ is between R and 1/R. Likewise, choose x 2 ∈ K 2 . There is a constant, still called R, such that the ratio of any γ ∈ Sim(M 2 ) satisfying γ(x 2 ) ∈ K 2 is between R and 1/R.

We assumed that Sim(M ) contains elements which are not isometries, so there exists γ 0 ∈ Sim(M 1 ) whose ratio is greater than R 3 . And since G • K = M , there exists γ = (γ 1 , γ 2 ) ∈ G such that γ(γ 0 (x 1 ), x 2 ) ∈ K. Then, γ 1 • γ 0 (x 1 ) ∈ K 1 , so the ratio of γ 1 • γ 0 is smaller than R, so the ratio of γ 1 is smaller than 1/R 2 . Meanwhile, γ 2 (x 2 ) ∈ K 2 , so the ratio of γ 2 is greater than 1/R. But since (γ 1 , γ 2 ) ∈ Sim(M ), γ 1 and γ 2 should have the same ratio, which is impossible. Consider a connected, compact manifold (M, ∇), where ∇ is a reducible, locally metric (but not globally metric) non-flat connection which preserves a conformal structure, and consider its universal cover M . Then π 1 (M ) is a subgroup of Sim( M ). Moreover, Sim( M ) contains elements which are not isometries. Thus, we may apply Proposition 9.25, which concludes the proof of Theorem 9.6. of f i 1 (x 0 ), which coincides with γ jip • γ • γ i 1 i near f i 1 (x 0 ), such that for all l ∈ {1, . . . , p -1}, d i l (γ i l-1 i l • • • • • γ i 1 i 2 (f i 1 (x 0 )), ∂U i l ) > � 0 , and γ i l i l+1 is defined on B g (γ i l-

1 i l • • • • • γ i 1 i 2 (f i 1 (x 0 )), � 0 ).
Since K n is saturated, the image of B g (f i 1 (x 0 ), � 1 ) by γ i l-1 i l • • • • • γ i 1 i 2 (when it is defined) does not intersect f i l (U i l ∩ K n ). Thus, this image is a ball of radius ≤ � 0 . By induction, this implies that this image is always well-defined, and thus, the ratio of γ is smaller than k 2 max � 0 /� 1 at f i (x 0 ) (where k max is still the maximum ratio of the γ ij ). By considering γ -1 , we see that the ratio is also bigger than � 1 /(k 2 max � 0 ). Thus, x 0 ∈ F , which concludes the proof.

Figure 1 .

 1 Figure 1.1 -A wooden realization of the Peaucellier straight-line motion linkage. Design: Adriane Kaïchouh, Mickaël Kourganoff, Thomas Letendre. Construction: Pierre Gallais.

Figure 1 . 2 -

 12 Figure 1.2 -The robotic arm R 3 . Recall that fixed vertices (here v 0 ) are represented by squares.

2 Figure 1 . 3 -

 213 Figure 1.3 -Thurston and Weeks' triple linkage. The three fixed vertices are on the unit circle and form an equilateral triangle, and there are two length parameters l 1 and l 2 .

Figure 1 . 4 -

 14 Figure 1.4 -The workspace of the central vertex in Thurston's triple linkage (in dark grey), for different choices of the lengths.

Figure 1 . 5 -

 15 Figure 1.5 -A centipede with 5 legs.

Figure 1 . 6 -

 16 Figure 1.6 -The pantograph. There is one rigid bar joining A (resp. C) to G, with a hinge D (resp. E) at the middle. The vertex A is fixed.

Figure 1 .Figure 1 2 -l 2 .

 1122 Figure 1.8 -Watt's linkage (obtained with Geogebra), with the workspace of the central vertex E. It has two fixed vertices A and B, and an additional vertex E at the middle of the edge CD. It is made of three bars AC, CD and DB: the bar CD has length 2c, while AC and DB have the same length length b. The distance between the fixed vertices A and B is 2a.

1. 2 . 3 Figure 1 .

 231 Figure 1.10 -The Peaucellier straight-line motion linkage. The distance between the two fixed points A and G is equal to s, the length of the edge GD.

In 1875 ,

 1875 Harry Hart discovered a new linkage for inversion, with only four bars (Figure 1.12). Similarly to the Peaucellier linkage, it is possible to add one fixed vertex and one edge to obtain Hart's straight-line motion linkage.

Figure 1 .Figure 1 .

 11 Figure 1.11 -A double-action steam engine 6 . Here, the straight-line motion linkage, used to guide the piston along a straight line, is not represented. The energy of the piston is transmitted to the wheel with only one bar.

  which gives the length of each edge 2 ); 2. A subset F ⊆ V of fixed vertices (represented by on the figures); 3. A function φ 0 : F → M which indicates where the vertices of F are fixed;

Proposition 3 . 3 .

 33 Let α, β ∈ M. If �α� + �β� = �α + β�, then α and β are colinear.

  In degenerate realizations, two vertices are sent to the same point of M. To avoid degenerate realizations, we add two vertices and five edges to the square abcd. We call this operation "rigidifying the square".

  so φ(e), φ(c) and φ(f ) are aligned, so φ(b), φ(c) and φ(d) are aligned and therefore φ(b)φ(a) = φ(c)φ(d).

Case 2 : 2 .

 22 φ(d) = φ(a)+φ(c) We have �φ(b)φ(a)� + �φ(b)φ(c)� = �φ(c)φ(a)� (= 2l), so φ(a), φ(c) and φ(b) are aligned, thus φ(d) = φ(b). We are taken back to the first case.

Case 3 :

 3 φ(d) � = φ(a)+φ(c) 2 and φ(a) � = φ(c). Let I = H(φ(a), l) ∩ H(φ(c), l). We have φ(d) ∈ I, φ(b) ∈ I, and card(I) ≤ 2. We have φ(a) + φ(c)φ(d) ∈ I. If φ(a) + φ(c)φ(d) = φ(d) then we are in the second case. If not, we have I = {φ(d), φ(a) + φ(c)φ(d)} and therefore, either φ

0 �

 0 Choose r, R, l > 0 and let F = {a}, φ 0 (a) = � 0 , P = {e}, Q = {d}. The square bdce is rigidified (see Section 3.2.2), but for convenience, we do not draw on the figure the vertices which are necessary for the rigidification. The vertex g and the two edges (bg) and (cg) are not part of the traditional Peaucellier inversor, but they are here to prevent φ(b) and φ(c) from being equal.

Figure 3 . 1 -

 31 Figure 3.1 -Two realizations of the same Peaucellier inversor in the Minkowski plane.

Fact 3 . 7 .

 37 The workspace of the vertex e, Conf P M (L), contains the spacelike cone� α ∈ M   �α� 2 > 0 � .Proof. Here, we use Proposition 3.2 at each step. Choose any α in the spacelike cone, let φ(e) = α, and choose φ(b) and φ(c) such that {φ(b), φ(c)} = H(φ 0 (a), ir)∩H(α, R). Then, �φ(c)φ(b)� 2 < 0, so it is possible to choose φ(d) such that H(φ(b), R) ∩ H(φ(c), R) = {φ(d), φ(e)}. Finally, choose φ(g) in the intersection H(φ(b), l) ∩ H(φ(c), il). Proposition 3.8. For this linkage, Reg P M (L) = Conf P M (L).

  L 2 : a robotic arm {a 2 , b 2 , e 2 } with one input e 2 and one fixed vertex a 2 , one edge {a 2 , b 2 } of length ir and one edge {b 2 , e 2 } of length R; L 3 : a rigidified square {b 3 , d 3 , c 3 , e 3 } with inputs b 3 , c 3 and four edges of length R; L 4 : a robotic arm {b 4 , g 4 , c 4 } with inputs b 4 , c 4 , one edge {b 4 , g 4 } of length l and one edge {g 4 , c 4 } of length il.

TakeF� 0 .5 t 0 �

 00 [START_REF][END_REF], to form the t 0 -integer linkage. The combination mappings β i send a, b, c and d respectively to the inputs e i of the linkages L i . = {a}; φ 0 (a) = ; P = ∅.

Figure 3 . 2 -

 32 Figure 3.2 -One realization of the diagonal parallelizer.

L 2 0

 20 be the linkage L 1 to which one adds new fixed vertices at �

0 � 0 �

 00 vertices a, b with φ 0 (a) = � 2 and φ 0 (b) = � π , but also two vertices c, d which are the inputs and correspond respectively to the variables x and y. Combine this linkage with a multiplier: the combination mapping β sends c to one of the inputs of the multiplier and d to the other one. The linkage (still called L) is now functional for (x, y) � -→ xy.

P 1 M

 1 (L 1 ) = B and remove the unnecessary inputs d 2n+2 , d 2n+4 . . . , d 2N . Then Conf P M (L) = A, which ends the proof Theorem 2.4.

Proposition 4 . 2 . 1 �

 421 This linkage is functional for the (Euclidean) inversion with respect to the circle C with hyperbolic center φ 0 (a) and hyperbolic radius arcosh cosh l cosh r . Proof. Let φ ∈ Conf H 2 (L). Let µ be the middle of the hyperbolic segment [φ(d)φ(e)]. First case. In this case, we assume φ 0 (a) = � 0 , x µ = 0 and y µ ≤ 1. φ(d) and φ(e) have two possible values each, and φ(d) � = φ(e) because t 1 � = t 2 . By symmetry, µ is also the middle of the hyperbolic segment [φ(b), φ(c)] and x φ(b) = x φ(c) = 0. If necessary, we exchange b and c so that y φ(c) ≥ y φ(b) . Let α be the Euclidean center of C. We have y α = y φ(a) cosh arcosh cosh l cosh r = cosh l cosh r . From the hyperbolic Pythagorean Theorem applied to the hyperbolic triangles (µφ(a)φ(d)), (µφ(b)φ(d)) and (µφ(c)φ(d)), letting D = δ(φ(d), µ), we get: δ(φ(a), µ) = arcosh cosh l cosh D δ(φ(c), µ) = δ(φ(b), µ) = arcosh cosh r cosh D We may now compute the coordinates of φ(b) and φ(c):

Figure 4 . 1 -Proposition 4 . 3 .

 4143 Figure 4.1 -The set K must not intersect the gray disk D or the black circle C. The following proposition tells us that these two obstructions are essentially the only ones. Proposition 4.3. Let C be a circle of hyperbolic center α ∈ H 2 , hyperbolic radius R > 0, Euclidean center µ and Euclidean radius r. Let D be the closed disk of hyperbolic center α and hyperbolic radius Q = δ(α, µ).Let K be a compact set in H 2 \ (C ∪ D). Then there exists a choice of φ 0 (a), l, r, t 1 , t 2 such that the Peaucellier linkage with these lengths is functional for the inversion with respect to C, and such that K ⊆ Reg P H 2 (L) ⊆ Conf P H 2 (L).

  , ψ(b), ψ(c) are on the same hyperbolic line � and

  3 , and L 5 = L 4 ∪ β L 3 . Rename the inputs: a := b 1 , b := b 2 , c := b 3 . Since the isometries of H 2 send the vertical line to other hyperbolic lines, the vertices a, b and c are always on the same hyperbolic line and L 5 is the desired linkage.

2

 2 Let P = {a, d, e}, F = ∅ and r > k 2 . Using a hyperbolic alignement linkage with parameter l = k 2 + 2r, force the three vertices a, b and c to move on the same hyperbolic line. In other words, combine the linkage on the figure above with a hyperbolic alignment linkage (the combination mapping sends a, b, c to the inputs of the hyperbolic alignment linkage).

  are now ready to construct our linkage. Start with a linkage L with the input set P = {d 2 , d 4 , . . . , d 2n } and no edge. Add other vertices a 1 , a 2 , . . . , a 2n , b 1 2

1 P

 1 = {a, b}, Q = ∅, F = ∅. The lengths of the edges l 1 and l 2 satisfy 0 < l 1 , l 2 ≤ π.

φ

  (j) � ∈ Rφ(m) because φ(m) ∈ φ(a) ⊥ and φ(j) � ∈ φ(a) ⊥ .

  is symmetric to φ(b) with respect to φ(a) ⊥ . Proof. Let us prove the first assertion. Let ψ ∈ (S 2 ) P , choose φ(g) anywhere in ψ(a) ⊥ ∩ ψ(b) ⊥ , and a point α ∈ φ(g) ⊥ ∩ φ(a) ⊥ . If δ(α, φ(b)) ≤ π/2, let φ(d) = α, else let φ(d) = -α. In any case we have δ(φ(d), φ(b)) ≤ π/2 so we can choose φ(e) on the intersection of the circles C(φ(b), π/4) and C(φ(d), π/4). Let φ(c) be symmetric to φ(b) with respect to φ(a) ⊥ , and let φ(f ) be the image of φ(e) by a half turn of axis Rφ(d). Then, φ is a realization of the linkage. We now prove the second assertion. Let φ ∈ Conf S 2 (L 2 ). If δ(φ(c), φ(d)) < π/2 then δ(-φ(c), φ(d)) > π/2 whereas δ(φ(b), φ(d)) ≤ π/2, so -φ(c) is not symmetric to φ(b) with respect to φ(a) ⊥ . If δ(φ(c), φ(d)) = π/2 then φ(c) = ±φ(a), which means that φ(c), φ(f ), φ(d), φ(e) are on the same geodesic and δ(φ(c), φ(e)) = 3π/4. Therefore φ(c) � = φ(b), so -φ(c) is not symmetric to φ(b) with respect to φ(a) ⊥ .

5. 2 . 1

 21 The homothety linkage Let λ ∈ (0, 1). Our aim is to construct a linkage which takes one point φ(a) = (x φ(a) , y φ(a) , 0) and, when possible, forces another point φ(b) = (x φ(b) , y φ(b) , 0) to satisfy y φ(b) = λy φ(a) .

  Our aim is to construct a linkage which takes two points φ(a) = (x φ(a) , y φ(a) , 0), φ(b) = (x φ(b) , y φ(b) , 0) and, when possible, forces a third point φ(c) = (x φ(c) , y φ(c) , 0) to satisfy y φ(c) = y φ(a) + y φ(b) .

1 . 2 . 4 .

 124 Restrict the two points a, b to move in the Oxy plane by using the fixed great circle linkage for k = 2. Using a symmetrizer, extend this linkage to a new one having a vertex d such that d is symmetric to b with respect to the plane {yz = 0}. Then for all φ ∈ Conf S 2 (L) we have y φ(b) = z φ(d) . 3. With two parallelizers, extend this linkage to a new one having a vertex e such that for all φ ∈ Conf S 2 (L): (a) y φ(e) = y φ(a) ; (b) z φ(e) = z φ(d) . With two parallelizers, extend this linkage to a new one with a vertex c such that for all φ ∈ Conf S 2 (L): (a) y φ(c) + z φ(c) = y φ(e) + z φ(e) ; (b) z φ(c) = 0.

Conf P S 2 ( 2 .

 22 L) = � ψ ∈ (S 2 ∩ (Oxy)) P   f (y ψ(a 1 ) , . . . , y ψ(a 3n ) ) = 0 � . With several symmetrizers, extend this linkage to a new one with vertices b 1 , b 4 , b 7 , . . . , b 3n-2 such that for all φ ∈ Conf S 2 (L) and for all k ∈ {1, . . . , n}:

2

 2 is the closure of an open set in B such that ∂D is a smooth manifold of dimension 1 without boundary: in other words, each component of ∂D is the image of a smooth embedding Γ : T 1 → B. Each curve Γ is called a wall of D: it has a unit tangent vector T and a unit normal vector N pointing toward Int D. The curvature of Γ is � dT ds   N �

Figure 6 . 1 -

 61 Figure 6.1 -The billiard reflection law.

Figure 6 . 2 -

 62 Figure 6.2 -A grazing collision on a dispersing billiard in T 2 . The flow stops being defined after this time.

Figure 6 . 3 -

 63 Figure 6.3 -On the left, a dispersing billiard in T 2 with infinite horizon. On the right, a dispersing billiard in T 2 with finite horizon.

  be a sequence of 2 × 2 matrices, with determinant ±1. Fix � > 0, and consider the cone C � of all vectors � x y � ∈ R 2 such that �y ≤ x ≤ 1 � y. Assume that for all k, and all v = � x y � with xy > 0,

Figure 6 . 5 -

 65 Figure 6.5 -Each A k maps the cone xy > 0 (in grey) into the smaller cone C � (in dark grey).

  ẏ = g(∇ ∂f ∂t ∂f ∂s , e) = g(∇ ∂f ∂s γ, e).

Figure 7 . 2 -

 72 Figure 7.2 -The projection of the geodesic onto the billiard (solid line) is close to the billiard trajectory (dotted line).

  δ→0

  lim �→0 |p y (t in )| = lim δ→0 lim �→0 |p y (t out )| .

Figure 7 . 3 -

 73 Figure 7.3 -The geodesic exits Z δ before a time which is O( √ δ), because p y (t)p y (0) is bounded from below by a small negative number.

Figure 8 . 1 -

 81 Figure 8.1 -A physical realization of our Anosov linkage.

1 rFigure 8 . 2 -

 182 Figure 8.2 -Mathematical description of our Anosov linkage.

Figure 8. 3 -

 3 Figure 8.3 -On the left, the configuration space of our linkage in T 2 × R, where T 2 is horizontal and R vertical. It is a surface of genus 7, with a self-intersection (at the center of the picture). On the right, the flattened configuration space, which is close to a billiard table with finite horizon (see also Figure 8.5).

Figure 8 . 4 -

 84 Figure 8.4 -The projection of the geodesics of the configuration space onto the billiard table. Theorem 7.1 states that the nongrazing geodesics are close to billiard trajectories. The behavior of grazing trajectories, described by Figure 7.1, is clearly visible here.

Fact 8. 4 .

 4 For all C 0 ∈ Conf(L) such that e � = 0 and e � = c, Conf(L) is locally a smooth graph above θ and φ near C 0 . More precisely, there exists a neighborhood U of C 0 in Conf(L), an open set V of T 2 and a smooth function F

Fact 8. 5 .

 5 a) For all C 0 ∈ Conf(L) such that (-cos θ -2, 0), (d, e) and (0, c) are not aligned, and such that φ � = 0 mod π, Conf(L) is locally a smooth graph above θ and c near C 0 . b) For all C 0 ∈ Conf(L) such that (cos φ + 2, 0), (d, e) and (0, c) are not aligned, and such that θ � = 0 mod π, Conf(L) is locally a smooth graph above φ and c near C 0 .

  and hence in H 2 ) as an algebraic set of R 2 which is contained in the half-plane. In fact, it turns out that the analogous definitions in the other usual models (the Poincaré disc model, the hyperboloid model, or the Beltrami-Klein model) are all equivalent. With this definition, we obtain the same results as in the Euclidean case: Theorem 2.6. Let A be a compact semi-algebraic subset of (H 2 ) n (identified with a subset of R 2n using the Poincaré half-plane model). Then, A is a partial configuration space of some linkage L in H 2 . When A is algebraic, one can choose L such that the restriction map Conf H 2 (L) → A is a smooth finite trivial covering.

  .2 Elementary linkages for geometric operations Euclidean circle. Conversely, if C is a Euclidean circle contained in the half-plane, it is also a hyperbolic circle, so there is a circle linkage such that Conf P H

4.2.1 The circle linkage

a b l We let F = {a} and P = {b} (see Section 3.1.1 for the notations). In this linkage Conf P H 2 (L) is a hyperbolic circle, which is also a 2 (L) = C. Moreover, Reg P H 2 (L) = Conf P H 2 (L).

  is close to 0. On the other hand, |p y (t)p y (t in )| remains bounded since the geodesic has unit speed, which concludes the proof of Statement (1) for t ∈ [t in , t 1 ]. To extend the result to t ∈ [t in , t in + √ δ], we prove that in fact t 1 = t in + √ δ: assume that t 1 � = t in + √ δ. Then p x (t) remains close to p x (t in ) for t ∈ [t in , t 1

  and Statement 1 is proved.

	√ For all t ∈ [0, 3	δ], we may write, as in the proof of Proposition 7.11: |p x (t) -p x (t in )| ≤ � sup B(q(t in ), 3 √ δ) � � � � N x N y � � � � t � � t in | ṗy | .
	Since Nx Ny does not depend on �,	�	sup B(q(t in ), 3 √	δ)	� � � Nx Ny	� � �	�	is close to 0. Moreover,
	� t t in	| ṗy | =	� t t in	�	ṗy + 2 ṗy	-�	≤ |p y (t) -p y (t in )| + 2	t in � t	ṗy	-.
	The term |p y (t) -p y (t in )| is bounded by 2, and which proves Statement 2.	� t t in ṗy	-is close to 0, so	� t t in | ṗy | is bounded,

On ne sait toujours pas, cependant, si l'on peut exiger ou non que l'espace de configuration soit difféomorphe à la variété M elle-même.

It is still unknown, however, whether it is possible to require that the configuration space be diffeomorphic to the manifold M itself.

This case corresponds, for example, to the condition (3/2 -|l1 -l2|)

+

3/4 < (l1 + l2) 2 < (3/2 + |l1 -l2|) 2 + 3/4 and |l1 -l2| < √ 3/2. 2In fact, we should show first that the configuration space is a smooth orientable surface, which is not so obvious, even for generic lengths l1 and l2. For example, one may compute directly the differential of the function F of Proposition 1.3.

These linkages seem to have no standard name in the literature.

Even for connected closed non-orientable surfaces!

c � User:Emoscopes / Wikimedia Commons / CC-BY-SA-3.0

c � User:Panther / Wikimedia Commons / CC-BY-SA-3.0

Our definition of semi-algebraic sets is not the standard one, but we know from the Tarski-Seidenberg theorem that the two definitions are equivalent (see[START_REF] Bochnak | Real algebraic geometry[END_REF]).

Of course, when M is a Riemannian manifold, we may choose all the lengths in R ≥0 !

already defined informally at the beginning of Section 2.3

We do not require U or π -1 W (U ) to be smooth manifolds: recall that a smooth map on π -1 W (U ) is, by definition, the restriction of a smooth map defined on the ambient M W ∪P .

Many authors change the topology of Ω in order to make the flow continuous, but it cannot be made differentiable.

It is not to be confused with the mixing rate of the billiard map, which we did not define here, and which is polynomial (see[START_REF] Chernov | Dispersing billiards with cusps: slow decay of correlations[END_REF]) in this case.

The word "cone" has several different meanings in mathematics: here we take the same definition as[START_REF] Katok | Introduction to the modern theory of dynamical systems[END_REF].

This terminology is common in the literature: see[START_REF] Victor | Using integrability to produce chaos: billiards with positive entropy[END_REF] or[START_REF] Wojtkowski | Two applications of Jacobi fields to the billiard ball problem[END_REF].

See the beginning of Chapter 6 for the definition of billiards.

In[START_REF] Burago | Uniform estimates on the number of collisions in semi-dispersing billiards[END_REF], Burago, Ferleger and Kononeko used such degenerate spaces (Alexandrov spaces) to estimate the number of collisions in some billiards.

These systems are called soft billiards in the literature: see also[START_REF] Bálint | Correlation decay in certain soft billiards[END_REF] for more details.

If the billiard trajectory has no bounce at all, then the geodesic remains outside of Z δ and the convergence is clear.

with the notations of Figure 1.3, l1 = � and l2 = 1

Remerciements

whence u(1) ≤ 2, which contradicts � -1 > 2 and u(1) ≥ � -1 . Thus, the assumptions of Theorem 6.8 are satisfied.

Theorem 6.8 provides us with a family of cones which satisfy invariance and expansion in the future. Moreover, φ t (x, v) = φ -1 t (x, -v), so there also exists a family of cones satisfying invariance and expansion in the past. By Proposition 6.7, the flow φ t has a stable and an unstable direction: thus, it is Anosov, and Theorem 6.3 is proved.

Proof of Theorem 6.4

In this proof, we will use the lemma: Lemma 6.13. Under the assumptions of Theorem 6.4, there exist m > 0 and t 0 > 0 such that every unit speed geodesic γ : [0, t 0 ] → M satisfies:

Proof. If the conclusion is false, consider a sequence (γ n ) of unit speed geodesics defined on [-n, n], such that for all n, � n

By the Arzelà-Ascoli theorem and a diagonal argument, one may extract a subsequence of γ n which converges uniformly on each [-n, n] to a geodesic defined on R. By dominated convergence, it satisfies � R K(γ(t)) = 0, which contradicts the assumption. Now, consider the values of m and t 0 given by lemma 6.13, and choose a geodesic γ. We may assume that m < 1 and, by dividing the metric of M by a constant if necessary, that t 0 < 1. Denote by u the solution of the Ricatti equation u � (t) = -K(t)u 2 (t) with u(0) = 0: if this solution is defined on [0,[START_REF] Laetitia | is flat[END_REF], let t 1 = 1; if not, write [0, t 1 ) the maximal interval on which the solution is defined. In particular, t 1 ≤ 1.

Set t 2 = sup {t ∈ [0, t 1 ] | u(t) ≥ m} (with t 2 = 0 if this set is empty). Thus, for all t ≥ t 2 , u � (t) = -K(t)u 2 (t) ≥ -m 2 .

If t 2 = 0, then for all t ∈ [0, t 1 ), using the estimate given by Lemma 6.13,

If t 2 � = 0, then for all t ∈ [t 2 , t 1 [, using the fact that K(t) ≥ 0,

In both cases, one gets u(t) ≥ mm 2 for all t ∈ [t 2 , t 1 [. Thus, the solution do not blow up to -∞, so t 1 = 1, and u(1) ≥ mm 2 . One may apply Theorem 6.3: the geodesic flow on M is Anosov and Theorem 6.4 is proved. Fact 8.6 shows that the metric g � is nondegenerate (although it is induced by a degenerate metric of T 2 × R 3 !), so with Fact 8.1 the physical behavior of the linkage is the geodesic flow on (Conf(L), g � ). Our aim is to show that it is an Anosov flow by applying Theorem 7.2.

Consider the projection onto the first coordinates:

Again with Fact 8.6, p| Conf(L) is an immersion: Conf(L) is isometric to a smooth surface Σ immersed in T 2 × R, endowed with the metric g � = dθ 2 + dφ 2 + � 2 dc 2 . We shall now call z the third coordinate instead of c, to be consistent with the notations of Theorem 7.2.

Denote by π : T 2 × R → T 2 the projection onto the first coordinates. The surface Σ projects to a smooth billiard table:

Its boundary has three connected components in T 2 : {cos θcos φ = 2r}, {cos θ + cos φ = 2r}, and {cos θ + cos φ = 2l -4}.

Figure 8.5 -The billiard table D (in grey) for r = 0.4 and l = 2.8. The billiard has negatively curved walls, which means that the obstacles are strictly convex.

There remains to show that the immersed surface Σ satisfies the 4 assumptions of Theorem 7.2. Assumption 1 is satisfied as a direct consequence of Fact 8.4. The following proposition proves Assumption 2.

Proposition 8.7. For all q ∈ π -1 (∂D) ∩ Σ, the curvature of Σ ∩ V is nonzero at q, where V is a neighborhood of q in the affine plane q + Vect(e z , (T q Σ) ⊥ ).

Chapter 9

Transverse similarity structures on foliations

Some background and vocabulary

In this section, we consider a connected manifold M endowed with an affine connection ∇ and recall some basic vocabulary. See [START_REF] Kobayashi | Foundations of differential geometry[END_REF] (for example) for more details about these notions.

Holonomy group. For x ∈ M , any continuous, piecewise C 1 , closed curve γ with base point x defines a linear map from the tangent space T x M to itself, given by parallel transport along γ. The set of all linear maps obtained in this way has a natural group structure: it is called the holonomy group at x, and written Hol x (∇).

For two base points x, y ∈ M , it is easy to see that the holonomy groups Hol x (∇) and Hol y (∇) are conjugated: thus, for most purposes, we may omit the base point and write simply Hol(∇).

The restricted holonomy group Hol 0 (∇) is the subgroup of Hol(∇) given by the contractible curves.

Irreducibility. We say that ∇ has irreducible holonomy (or simply, M is irreducible) if, for x ∈ M , the only subspaces of T x M stable under Hol x (∇) are T x M and {0}. Otherwise, we say that ∇ is reducible. If the property holds for Hol 0

x (∇) instead of Hol x (∇), we say that (M, ∇) is locally irreducible or locally reducible.

Connections preserving structures. Given a Riemannian metric g on M , we say that ∇ preserves g if for any curve γ joining x ∈ M to y ∈ M , the parallel transport P γ : T x M → T y M is an isometry for the metric g.

Likewise, for a conformal structure c on M , we say that ∇ preserves c if for any curve γ joining x ∈ M to y ∈ M , the parallel transport P γ : T x M → T y M is a conformal map.

Introduction

Decomposition of locally metric connections

We recall the famous decomposition theorem of De Rham [dR52]:

where k ≥ 0, M 0 is flat, and M 1 , . . . , M k are non-flat, locally irreducible manifolds. Moreover, π 1 (M ) acts on M as a subgroup of

This decomposition is unique up to the order of the factors.

Proof. Choose any metric g preserved by ∇. In the decomposition given by Theorem 9.3, the metric g induces metrics g 0 , g 1 , . . . , g k on M 0 , M 1 , . . . , M k . For i ∈ {1, . . . , k}, since ∇ i is irreducible, Fact 9.13 implies that Aff(M i , g i ) = Sim(M i , g i ).

Transverse similarity structures

The main tool in the proofs of Theorems 9.3 and 9.6 is the study of transverse similarity structures on foliations. Such foliations may be seen as a particular case of (transversally) conformal foliations, or a generalization of (transversally) Riemannian foliations.

Let (M, F) be a compact foliated manifold, and (U i ) 1≤i≤r a covering of M compatible with the foliation, such that each U i is diffeomorphic to V i × T i , where V i (the plaque) is an open ball of R p and T i an open ball of R q . We denote by f i : U i → T i the projections, T = ∪ 1≤i≤r T i the (global) transversal, and define the transition maps (γ ij ) i,j :

The pseudogroup Γ spanned by the (γ ij ) is called the holonomy pseudogroup of the foliation.

A transverse similarity structure on the foliation F is a metric g on the transversal T such that the transition maps γ ij are local similarities (i.e. belong to Sim loc (T )). The foliation is said to be transversally Riemannian (or simply Riemannian) if it is possible to choose g such that the γ ij are isometries.

Our main result on transverse similarity structures is the following:

Theorem 9.15. Let (M, F) be a compact foliated manifold with a transverse similarity structure. Then one of the following two facts occurs:

1. The transverse similarity structure on the foliation F is flat ( i.e. the metric g on the transversal T is flat);

2. The foliation F is transversally Riemannian ( i.e. there exists a metric h on the transversal T such that the transition maps are isometries).

We prove Theorem 9.15 in Section 9.3, and give two alternative proofs in Appendix B, one of which is valid only in the analytic case.

Notice that Theorem 9.15 does not assume that M is endowed with a locally metric connection, but only that there is a transverse similarity structure on the foliation: the setting is more general than for Theorem 9.3.

About the foliated Ferrand-Obata conjecture. For transversally conformal foliations, there is an analogue of Theorem 9.15 (see [START_REF] Tarquini | Feuilletages conformes[END_REF]): Theorem 9.16 (Tarquini, 2004). Any transversally analytic conformal foliation of codimension ≥ 3, on a compact connected manifold, is either transversally Möbius or Riemannian.

It is also believed that Theorem 9.16 should be valid without the analyticity assumption: this is the foliated Ferrand-Obata conjecture. Our Theorem 9.15 implies the following:

Corollary 9.17. The foliated Ferrand-Obata conjecture is true if the transverse conformal structure on the foliation is induced by a transverse similarity structure.

Foliations with transverse similarity structures

In this section, we prove Theorem 9.15. We refer to Section 9.2.4 for the basic notations.

A foliation is said to be equicontinuous if its holonomy pseudogroup Γ is equicontinuous. If the foliation has a transverse similarity structure, equicontinuity is equivalent to the existence of a constant m > 1 such that the ratio of any γ ∈ Γ at any

The following proposition is crucial in the proof of Theorem 9.15:

Proposition 9.18. Any equicontinuous foliation with a transverse similarity structure is Riemannian.

Proof. This theorem is proved in [START_REF] Tarquini | Feuilletages de type fini compact[END_REF]: more generally, any foliation with compact finite type is Riemannian. Now, our first step in the proof is based on a trick which was described in [START_REF] Frances | Autour du théorème de Ferrand-Obata[END_REF].

Proposition 9.19. Let (M, g) be a Riemannian manifold whose Riemann tensor R does not vanish. Then Sim loc (M ) preserves a Riemannian metric.

Proof. If R denotes the Riemann tensor, define �R� g (x) as the supremum of the values of �R x (u, v)w� g when u, v, w are vectors of T x M which have unit length for g. Then the metric �R� g g is invariant by Sim loc (M ).

Thus, if (M, F) is a foliated manifold with a transverse similarity structure, either F is Riemannian, or the Riemann tensor of (T, g) vanishes somewhere. Our aim is to show that, in the last case, the Riemann tensor vanishes in fact everywhere.

Another useful tool to prove flatness is the following:

Proposition 9.20. Consider a connected Riemannian manifold (M, g), an open subset U ⊆ M and a similarity φ ∈ Sim loc (M ), φ : M → U . Assume that φ has a fixed point x ∈ M , and that its ratio is r φ < 1. Then:

Since φ n (y) tends to the fixed point x, the quantity �R� g (φ n (y)) is bounded. Thus, �R(a, b)c | d� = 0 and the first statement is proved.

For the second statement, notice that since M is flat, the exponential map exp x : B(0, �) → B g (x, �) is an isometry for some � > 0 (where B(0, �) is the ball in T x M of center 0 and radius � for the Euclidean metric g x , while B g (x, �) is the ball in M of center x and radius � for the distance induced by g).

Thus, for all n ≥ 0,

). Since φ n preserves the Levi-Civita connection of g, we have

Until the end of this section, we consider a compact, connected foliated manifold (M, F) with a transverse similarity structure. We still denote by g a metric on the transversal T such that the γ ij are local similarities. This metric g induces a distance d i on each T i . Lemma 9.21. There exists � 0 > 0 such that for all x ∈ M , there exists i ∈ {1, . . . , r} which satisfies x ∈ U i and d i (f i (x), ∂T i ) > � 0 (see Section 9.2.4 for the notations).

Proof. Assume the contrary: there exists a sequence (x n ) n∈N in M such that for all i ∈ {1, . . . , r} with x n ∈ U i , we have d i (f i (x n ), ∂T i ) ≤ 1/n. Since M is closed, we may assume that x n converges to some x ∞ ∈ M . Then x ∞ is in some U i 0 , and for any large enough n, x n ∈ U i 0 . Hence,

In the following, we fix this � 0 .

Lemma 9.22. Let γ ∈ Γ, x ∈ M and i ∈ {1, . . . , r}, such that γ is defined on a neighborhood of f i (x) in T i and takes its values in T j .

Then there exists γ =

Proof. It results from the general theory of holonomy pseudogroups that γ is obtained by following some curve c : [0, 1] → M such that c(0) = x, and f j (c(1)) = γ(f i (x)). For each i ∈ {1, . . . , r}, we define E i as the set of all open intervals (a, b) such that for all t ∈ (a, b), c(t) ∈ U i and d i (f i (c(t)), ∂T i ) > � 0 . Lemma 9.21 implies that ∪ 1≤i≤r E i is an open cover of [0,[START_REF] Laetitia | is flat[END_REF]: it has a finite subcover {(a 1 , b 1 ), . . . , (a p , b p )}, to which corresponds a sequence i 1 , . . . , i p of indices. Then, γ coincides with

Lemma 9.23. Let E be the set of all x ∈ M for which there exists m > 1 such that for all i ∈ {1, . . . , r} with x ∈ U i , every γ ∈ Γ defined on f i (x) has ratio ≥ 1/m at f i (x).

1. In the definition of E, it is possible to choose m independently of x.

2. If E is non-empty, then E = M and Γ is equicontinuous.

Proof. We start with the proof of the first statement. Assume that there is no uniform bound: then, there exist sequences (x n ), (i n ), (j n ) and (γ n ) such that x n ∈ E, γ n is defined on a neighborhood of f i n (x n ) in T i n , takes its values in T j n , and the ratio of γ n is ≤ 1/n at f i n (x n ). Let k max be the maximum ratio of γ ij for i, j ∈ {1, . . . , r}.

The natural idea is that the points x n accumulate somewhere because M is compact: if all the terms γ n were defined on balls of the same radius � 0 , then an infinity of them would be well-defined on the same term x n 0 , which would contradict the fact that x n 0 ∈ E. In fact, the domain of γ n can be arbitrarily small, so we need to consider another sequence (y n ) instead of (x n ).

For each n, Proposition 9.22 gives us a γn =

and the ratio of

Choose q n ∈ {1, . . . , p n -1} which minimizes the ratio of

Thus, ρn is well-defined on B g (f i n q n (y n ), � 0 ) and has ratio ≤ k 2 max /n at f i n q n (y n ). Since M is compact, we may assume up to extraction that (y n ) converges to a limit y ∈ M (and y ∈ U i for some i): There exists n 0 > 0 such that for all n ≥ n 0 , y n ∈ U i and f i (y n )f i (y) < � 0 /(3k max ). Thus, ρn is well-defined on f i n 0 q n 0 (y n 0 ) for all n ≥ n 0 , which contradicts the fact that y n 0 ∈ E and ends the proof of the first statement.

To prove the second statement, first notice that for all x ∈ E, and all i ∈ {1, . . . , r} such that x ∈ U i , every γ ∈ Γ defined on f i (x) (taking values in T j ) has ratio ≤ m at f i (x): otherwise, γ -1 would have ratio < 1/m at f j (x), which contradicts the fact that γ(f i (x)) ∈ f j (E).

Since M is connected, it suffices to show that E is open and closed in M . Thus, Γ will be equicontinuous on M .

Let us show that E is open. Let x 0 ∈ E and i such that

Let us show that V ⊆ E: let y 0 ∈ V , i 0 ∈ {1, . . . , r}, and γ ∈ Γ defined on f i 0 (y 0 ), taking its values in T j 0 .

With Lemma 9.22, there exists a γ =

In particular, by induction on l, γ i l i l+1 is well-defined on a ball centered at

In this section, we adapt the ideas of the proof of De Rham's theorem available in [START_REF] Kobayashi | Foundations of differential geometry[END_REF], to give a direct proof of Theorem 9.4.

Consider a connected, simply connected Riemannian manifold M with reducible holonomy, and write T x (M ) = E �

x ⊕ E �� x a decomposition into subspaces invariant by holonomy (of dimension at least 1). Denoting by E � and E �� the corresponding distributions, define M � (x) and M �� (x) the maximal integral manifolds of E � and E �� through x, for any x ∈ M .

The local decomposition theorem of De Rham states that such a manifold is locally a product: there is a neighborhood of x which is isometric to a product of Riemannian manifolds V � × V �� . But for the global version of the theorem (Theorem 9.1), one has to assume, classically, that M is complete: here, we do not make this assumption. Definition 9.26. Let x ∈ M and X(t) a curve in T x M . The development of X(t) into M , when it exists, is the (unique) curve φ starting from x, such that X(t) is the result of the parallel displacement of dφ dt (t) along the curve φ| [0,t] itself.

Proposition 9.27. The development of X(t) into M is always unique. Moreover, if M is complete, it always exists.

Proof. It is the solution of a differential equation. For more details, see [START_REF] Kobayashi | Foundations of differential geometry[END_REF], Chapter IV, Theorem 4.1.

Definition 9.28. For two curves φ, ψ : [0, 1] → M with φ(1) = ψ(0), we denote by ψ • φ : [0, 1] → M the concatenation of the two curves, i.e.

In the same spirit, we define φ -1 so that φ -1 (t) = φ(1t).

Lemma 9.29. In this lemma, we assume that either M � (x) is complete for all x ∈ M , or M �� (x) is complete for all x ∈ M .

Let φ : [0, 1] → M be a curve tangent to E � and ψ : [0, 1] → M a curve tangent to E �� , such that φ(0) = ψ(0). Then there exists a unique smooth homotopy f :

Proof. By symmetry, we may assume that M � (x) is complete for all x ∈ M .

Let s ∈ [0, 1]. For all t, let X(t) be the result of the parallel displacement of dφ dt (t) along the curve φ| [0,t] • ψ| -1 [0,s] . From properties 1, 2, 3 and 5, we deduce that f (., s), if it exists, is necessarily the development of X(t) into M � (ψ(s)), so we have unicity. For the existence, we must check that f defined in this way (for all s ∈ [0, 1]) satisfies all of the desired properties.

First, consider t 0 and s 0 such that [0, t 0 ] × [0, s 0 ] lies in a open set which is isometric to a product V � × V �� (obtained with the local version of De Rham's theorem). Then it is clear that Properties 3, 4 and 5 are satisfied for all (t, s)

Applying Properties 3, 4 and 5 to each V i one after another, one proves that these properties are satisfied for (t, s)

Let s 1 be the upper bound of all s 0 such that f satisfies properties 3, 4 and 5 for all (t, s) ∈ [0, 1] × [0, s 0 ]. Notice that the three properties are satisfied for s 1 itself, by taking the limit when s → s 1 , s < s 1 , in each property. Then, if s 0 < 1, we may cover f (., s 1 ) by a finite number of open sets of the form V � × V �� and prove, as above, that the three properties are satisfied for (t, s) ∈ [0, 1] × [s 1 + �] with � > 0, which contradicts the definition of s 1 . Thus, s 1 = 1 and the lemma is proved. Definition 9.30. Let φ : [0, 1] → M be any curve, and write x = φ(0). Let X(t) be the result of the parallel displacement of dφ dt (t) along the curve φ| [0,t] itself. Let X � (t) and X �� (t) be the projections of X(t) on E �

x (M ) and E �� x (M ). We define the projection of φ onto

In view of Proposition 9.27, the projection onto M � (x) is always well-defined because we assumed that M � (x) is complete, but it is not the case for the projection onto M �� (x).

Lemma 9.31. Let x, y ∈ M and φ : [0, 1] → M joining x to y.

1. The projection of φ onto M � (x) and the projection of φ -1 onto M �� (y) are welldefined. We call them τ � and τ �� .

2. The curves τ � and τ �� have the same endpoint.

3. This endpoint depends only on x and y (but not on the curve φ). It is called the projection of y onto M � (x). The projection of y onto M �� (x) is defined in the same way, by exchanging the roles of the distributions E � and E �� .

Each curve φ i joins two points, which we name x i and x i+1 .

Consider φ � r the projection of φ r onto M � (x r ), and let φ �� r be projection of φ -1 r onto M �� (x r+1 ). Since V r is a product, it is clear that the projections are well-defined and have the same endpoint. Let τ � r = φ � r and τ �� r = φ �� r . Continue the construction of φ � k , φ �� k , τ � k and τ �� k by induction for k = r -1 to 1: define φ � k as the projection of φ k onto M � (x k ), and φ �� k as the projection of φ -1 k onto M �� (x k+1 ). Notice that the parallel displacement along φ k is the same as the parallel displacement along (φ �� k ) -1 • φ � k . Then, construct a homotopy f (t, s) using the curves φ �� k (s) and τ � k+1 (t), and Lemma 9.29. Let

k is the projection of (φ r • . . . • φ k ) -1 onto M �� (y). By construction, τ � k and τ �� k have the same endpoint. Thus, assertions 1 and 2 are proved by applying this result for k = 1. Now, we prove that the endpoint of τ � depends only on x and y, i.e. does not depend on the choice of φ. Consider another curve ψ joining x to y: since M is simply connected, there is a homotopy between the two curves. Moreover, we may assume that this homotopy is "small", i.e. there exists j ∈ {0, . . . , r} and ψ j : [0, 1] → V j joining x j to x j+1 such that

Make the above construction for ψ. The point is that the parallel transport of E � -vectors along ψ �� j between x j+1 and ψ �� j (1) is the same as the parallel transport along φ �� j (because ψ �� j and φ �� j are both E �� -curves joining the same points). Thus, for each k, the endpoint of τ � k does not change if we consider the curve ψ instead of φ.

Lemma 9.32. Let x, y ∈ M . Denote by y � and y �� the projections of y onto M � (x) and M �� (x). Then y is the projection of y � onto M � (y �� ), as well as the projection of y �� onto M �� (y � ).

Proof. Consider a curve φ joining x to y, and its projections τ � and τ �� , as in Lemma 9.31. Also write α �� the projection of φ onto M �� (x), and α � the projection of φ -1 onto M � (y).

By Lemma 9.31, the projections of τ and (α � ) -1 onto M � (y �� ) have the same endpoint. Since the projection of (α � ) -1 is (α � ) -1 itself, this endpoint is y. Thus, the endpoint of the projection of τ � • (α �� ) -1 onto M � (y �� ) is also y. This means exactly that the projection of y � onto M � (y �� ) is y.

End of the proof of Theorem 9.4. Let x 0 ∈ M , and define Φ : M → M � (x 0 ) × M �� (x 0 ), which to a point y ∈ M associates its projections onto M � (x 0 ) and M �� (x 0 ). The product M � (x 0 ) × M �� (x 0 ) is naturally endowed with the product of the metrics induced by M . We claim that Φ is an isometry between M and M � (x 0 ) × M �� (x 0 ).

Injectivity is a direct consequence of Lemma 9.32. For surjectivity, consider any (y � , y �� ) ∈ M � (x 0 ) × M �� (x 0 ), and let y be the projection of y � onto M � (y �� ). Since x 0 is the projection of y � onto M �� (y �� ), it results from Lemma 9.32 that the projection of y onto M � (x 0 ) is y � . Symmetrically, the projection of y onto M �� (x) is y �� , and thus Φ(y) = (y � , y �� ).

The only thing left to do is to show that Φ is isometric at each point. Consider y ∈ M and v ∈ T y M , with the decomposition

. Thus, v � has the same length as d dt � � t=1 τ , which has itself the same length as Φ * (v � ). The same result applies to v �� , so Φ is an isometry, and Theorem 9.4 is proved.

Appendix B. Two other proofs of Theorem 9.15

We present two alternative proofs of Theorem 9.15. We keep the notations introduced at the beginning of Section 9.3.

B1. A simpler proof in the analytic framework

This proof uses Proposition 9.18 and Lemma 9.22.

Assume that (T, g) is not Riemannian: by Proposition 9.18, it is not equicontinuous, so for all m > 0, there exists γ = γ i p-1 ip • . . . • γ i 1 i 2 ∈ Γ with ratio ≤ m at some f i 1 (x), x ∈ M . With Lemma 9.22, we may assume that for all l ∈ {1, . . . , p -1},

Choose � ∈ (0, � 0 ) such that the injectivity radius of the exponential map of (T, g) is at least � at every point x ∈ T satisfying d

If m is chosen small enough, there exist s, t with 1 ≤ s ≤ t ≤ p such that:

Let γ 0 = γ i t-1 it • . . . • γ isi s+1 . In the analytic framework, there is a canonical way to extend the domain of γ 0 to the closed ball B = B g (γ i l-1 i l • • • • • γ i 1 i 2 (f it (x)), �) so that γ 0 remains a similarity of ratio ≤ 1/2, using the exponential map of g: the new γ 0 maps the ball B into itself, and thus it has a fixed point. Therefore, the transversal T is flat on B by Proposition 9.20, and again by analyticity, it is flat everywhere, which concludes the proof.

B2. Another proof in the C ∞ framework

This proof uses Proposition 9.18, Proposition 9.19 and Lemma 9.22.

The equicontinuity domain F is defined as the set of all x ∈ M for which there exists m > 1 such that for all i ∈ {1, . . . , r} with x ∈ U i , every γ ∈ Γ defined on f i (x) has ratio between 1/m and m at f i (x). Assume that the foliation is not transversally flat: then the set F is non-empty by Proposition 9.19. We want to prove that F = M .

It is possible to show that F is open, by copying the proof that E is open in Lemma 9.23: we leave this to the reader.

Let (K n ) be an exhaustion of F by compact sets, i.e. an increasing sequence of compact subsets of F whose union is F . Since F is saturated (i.e. it is a union of leaves), it is possible to assume that every set K n is itself saturated (take the closure of the union of all leaves which intersect K n : this new set is saturated, since it is the closure of a saturated set).

Choose n so large that for all i ∈ {1, . . . , r} and all x ∈ U i ∩ ∂F such that d i (x, ∂U i ) ≥ � 0 , the ball B g (f i (x), � 0 ) intersects f i (K n ∩ U i ). Then there is also an � 1 > 0 such that for all i ∈ {1, . . . , r} and all x ∈ U i ∩ ∂F , the ball B g (f i (x), � 1 ) does not intersect f i (K n ∩ U i ).

Assuming that F � = M , we choose x 0 on ∂F (the boundary of F ) and look for a contradiction. Notice that the whole leaf of x 0 is contained in ∂F , because F is saturated. The idea is that, under the action of the holonomy group, the images of a transverse ball centered at x 0 remain small because their center is on ∂F , while they cannot intersect K n : thus x 0 is in the equicontinuity domain F , which contradicts the fact that F is open. The details now follow.

Consider i ∈ {1, . . . , r} such that x 0 ∈ U i and γ ∈ Γ defined on a neighborhood of f i (x 0 ). With Lemma 9.22, there exists a γ = γ i p-1 ip • • • • • γ i 1 i 2 defined on a neighborhood