major areas with decorated antler artefacts graves with antler structures debitage shaping

Deer Antler during the French Mesolithic Technical Exploitation and Symbolic Aspects

Bibliography :

(1) Ginestet, J.-P., Bintz, P., Chaix, L., Evin, J. et Olive, C. (1984). L'abri sous roche de la vieille église La Balme-de-Thuy (Haute Savoie) Premiers Résultats. Bulletin de la Société Préhistorique Française, t.81, n °10-12, p. 320-342. In some graves, deer antlers (often complete, from hunted animals or shed antlers) are associated with the bodies, forming tent-like structures (Téviec) (fig. 9) or frames (Hoëdic) (fig. 8) over the head and shoulders.

Such graves at Téviec (2 examples) and Hoëdic

(4 examples) tend to be of the oldest individuals and the richest burials. M. and S.-J. Péquart discussed the significance of these structures (from hunting trophies to the burial of shamans) and highlighted their prestige nature. A recent study has noted the link between antler structures and the richness ob object placed with the body (11).

Antler exploitation at the Cuzoul de Gramat Antler use at Teviec and Hoedic

At Teviec and Hoëdic, complete antlers were placed in the graves without major modification. Was the shape suggestive ? Was it a way to represent the whole deer, a major game animal ? But in that case, why use shed antler ? Or were these deciduous hard material a regeneration/rebirth symbol ? In any case, the link between antler structures and prestige is clear. The Mesolithic populations seemed to desire the antler for its symbolic aspects rather than for its mechanical properties.

Mesolithic populations exploited red deer antler mainly to produce heavy bevelled tools used as mattocks or picks rather than as axes as suggested by use-wear traces (12). Antler is a perfect raw material being both hard and flexible and so able to resist shock without breaking. Mesolithic populations were more concerned with the physical and mechanical properties of antler.

Appearance was a minor importance: shaping was limited to the active part, while debitage marks were not removed by the craftsmen and there was no finishing or decoration of the final object. New study of the former collections of Cuzoul de Gramat enabled us to reconstruct precisely the technical scheme for the transformation of deer antler. This scheme, or more often parts of it, has been identified identified in material from sites throughout France (for exemple (1), (4)...). However, the sites of Teviec and Hoëdic are exeptionnal for the region while the use of antler structures in funerary deposits are also limited to these two sites. Teeth : mainly canines (perforated and used as ornaments) Bones : mainly long ones (often exploited with simple actions, like percussion) Antlers (fig. 1) : used to make domestic tools but also symbolic objects Red deer was one of the main game animal of the french Mesolithic. A source of food, it was also the source of various raw materials :

Emblematic raw material of Azilian cultures (flat harpoons) and, later, of the Neolithic (haft sockets), the exploitation of deer antler during the French Mesolithic is less well known but some recent studies have begun to shed light on the production of antler tools. We shall expose the use of antler via three French examples: one a domestic context (recently studied from a technological point of view) and two funerary contexts, all dated to 2nd Mesolithic. We shall show that the same raw material was used for very different purposes, behaviours that are also seen beyond France. If we compare the distribution of decorated tools and antler funerary structures on larger European scale (fig. 12), the division between the northern third part and the rest of France can be seen in the light of northern European influences.

Furthermore, in the case of tool production, antler debitage is also different between northern and southern Europe. In northern countries, base, including burr, is a part of the finished objects (and sometimes the active part). In the south, the base is a waste product (fig. 13).

Deer antler during the french Mesolithic :

Fig. 2 :

 2 Fig. 2 : annual evolution of red-deer antlers

Fig. 3 :

 3 Fig. 3 : stratigraphic section. 2nd Mesolithic levels are framed (from (3))

Fig

  Fig. 10 : decorated antler shaft from Longueil, France (10)

Fig. 4 :

 4 Fig. 4 : technical transformation scheme of deer antler

N

  . Cavanhié, A. Chevalier, J.B. Fourvel, D. Kuntz, J. Lacarrière, M. Loughton, M.C. Soulier : thank you ! a same raw material for very different purposes throughout France, importance of physical and mechanical properties of antler in the third north, importance of symbolic aspects beware of the problem of differential preservation northern european influences ? > decoration / funerary structures / debitage bog sites in the north Vs cave sites in the south hard to compare antler exploitation on European scale


Evelyne Hubert

Evelyne Hubert Generation

Generation properties of Maurer-Cartan invariants

come

Introduction

Group actions are ubiquitous in mathematics and arise in diverse fields of science and engineering. Whether algebraic or differential, one can distinguish two families of applications for invariants of group actions: equivalence problems and symmetry reduction. In equivalence problems, invariants come as functions whose values separate the orbits and thus distinguish inequivalent problems. Symmetry reduction postulates that invariants are the best coordinates in which to think a problem. Here a generating set of invariants is needed, so as to rewrite the problem in terms of those. In this paper we show that the set of differential invariants that has classically been used for equivalence problems in differential geometry is also a generating set, endowed with a simple rewriting algorithm.

In differential geometry equivalence problems are diverse though their resolutions often take their roots in the work of Elie Cartan [START_REF] Cartan | La méthode du repère mobile, la théorie des groupes continus, et les espaces généralisés[END_REF][START_REF] Cartan | La théorie des groupes finis et continus et la géométrie différentielle traitées par la méthode du repère mobile[END_REF]. For the equivalence of submanifolds in a homogeneous space, an interpretation of Cartan's moving frame method [START_REF] Griffiths | On cartan's method of lie groups as applied to uniqueness and existence questions in differential geometry[END_REF][START_REF] Gary | Higher order contact of submanifolds of homogeneous spaces[END_REF][START_REF] Clelland | Lecture notes from the MSRI workshop on Lie groups and the method of moving frames[END_REF][START_REF] Ivey | Cartan for beginners: differential geometry via moving frames and exterior differential systems[END_REF] exhibits separating invariants: they are are the coefficients of the pull-back of the Maurer-Cartan forms to the submanifold and we shall call those the Maurer-Cartan invariants. The finite generation of differential invariants, for the more general case of pseudo-groups, was addressed in [START_REF] Tresse | Sur les invariants des groupes continus de transformations[END_REF][START_REF] Kumpera | Invariants différentiels d'un pseudogroupe de Lie[END_REF][START_REF] Kumpera | Invariants différentiels d'un pseudogroupe de Lie[END_REF][START_REF] Kumpera | Invariants différentiels d'un pseudogroupe de Lie[END_REF][START_REF] Muñoz | On the finiteness of differential invariants[END_REF] -see also [START_REF] Ovsiannikov | Group analysis of differential equations[END_REF][START_REF] Olver | Equivalence, invariants and Symmetry[END_REF] for Lie groups. In their reinterpretation of Cartan's moving frame method, Fels and Olver [START_REF] Fels | Moving coframes. II. Regularization and theoretical foundations[END_REF][START_REF] Mansfield | Algorithms for symmetric differential systems[END_REF][START_REF] Kogan | Invariant Euler-Lagrange equations and the invariant variational bicomplex[END_REF] addressed equivalence problems as well as finite generation. Normalized invariants are the focus there. They form a generating set and the relationships (syzygies) they satisfy were given in [START_REF] Hubert | Differential invariants of a Lie group action: syzygies on a generating set[END_REF].

In this paper we show that Maurer-Cartan invariants form a generating set. On one hand we shall give the explicit formulae for the Maurer-Cartan invariants in terms of the normalized invariants. Besides the methods in [START_REF] Fels | Moving coframes. II. Regularization and theoretical foundations[END_REF][START_REF] Kogan | Two algorithms for a moving frame construction[END_REF], there is an algorithm for computing normalized invariants in the algebraic case [START_REF] Hubert | Rational invariants of a group action. construction and rewriting[END_REF][START_REF] Hubert | Smooth and algebraic invariants of a group action. Local and global constructions[END_REF]. On the other hand we show how to rewrite any differential invariant in terms of the Maurer-Cartan invariants. This is a simple recursive process.

The generation property is a rather simple observation, yet meaningful and unifying. We draw a link between the reinterpretation of the moving frame by Griffiths [START_REF] Griffiths | On cartan's method of lie groups as applied to uniqueness and existence questions in differential geometry[END_REF][START_REF] Gary | Higher order contact of submanifolds of homogeneous spaces[END_REF][START_REF] Clelland | Lecture notes from the MSRI workshop on Lie groups and the method of moving frames[END_REF][START_REF] Ivey | Cartan for beginners: differential geometry via moving frames and exterior differential systems[END_REF] on one hand and Fels and Olver [START_REF] Fels | Moving coframes. II. Regularization and theoretical foundations[END_REF][START_REF] Kogan | Invariant Euler-Lagrange equations and the invariant variational bicomplex[END_REF][START_REF] Hubert | Rational invariants of a group action. construction and rewriting[END_REF][START_REF] Hubert | Differential invariants of a Lie group action: syzygies on a generating set[END_REF][START_REF] Mansfield | Invariant Calculus for Differential and Discrete Problems[END_REF] on the other hand. We also show how it links with the invariants arising in curve evolution [START_REF] Beffa | The theory of differential invariants and KdV Hamiltonian evolutions[END_REF][START_REF] Beffa | Relative and absolute differential invariants for conformal curves[END_REF][START_REF] Beffa | Poisson geometry of differential invariants of curves in some nonsemisimple homogeneous spaces[END_REF][START_REF] Beffa | On completely integrable geometric evolutions of curves of lagrangian planes[END_REF][START_REF] Beffa | Projective-type differential invariants and geometric curve evolutions of kdv-type in flat homogeneous manifolds[END_REF][START_REF] Mansfield | Evolution of curvature invariants and lifting integrability[END_REF].

Beside their geometrical significance, Maurer-Cartan invariants have a number of desirable features over normalized invariants. First their number is bounded by the product of the submanifold dimension with the dimension of the group. Second, their syzygies can be written down explicitly from the structure equations of the group. Then, a meaningful item for symbolic computation, they allow to avoid denominators in the symmetry reduction.

In Section 1 we describe part of the moving frame method as reinterpreted by Griffiths to show the prominent role of Maurer-Cartan forms, and their pull-backs, have held for equivalence problems in differential geometry. The geometrical construction of normalized invariants based on a cross-section is reviewed in Section 2. The Maurer-Cartan invariants come in Section 3. They arise as the coefficients of the pullback of the Maurer-Cartan forms and define the difference between the invariantization of the total derivation and the invariant derivation of the invariantization. It is from this latter formulae that one deduces in Section 4 that the Maurer-Cartan invariants, along with the normalized and edge invariants, form a generating set. The differential relationships they satisfy are discussed in Section 5. In Section 6 we show how to recast our results in the context of the better known case of matrix groups. In place of running examples along the text, we have gathered those in Section 7.

Equivalence of submanifolds

The content of this section is rather independent from the rest of the paper. Its purpose is to link our approach to the interpretation of the moving frame method in [START_REF] Griffiths | On cartan's method of lie groups as applied to uniqueness and existence questions in differential geometry[END_REF][START_REF] Gary | Higher order contact of submanifolds of homogeneous spaces[END_REF][START_REF] Clelland | Lecture notes from the MSRI workshop on Lie groups and the method of moving frames[END_REF][START_REF] Ivey | Cartan for beginners: differential geometry via moving frames and exterior differential systems[END_REF]. We show where the Maurer-Cartan invariants occur.

We consider a r-dimensional Lie group G. We make use of a basis of invariant vector fields on G, v = (v 1 , . . . , v r ). They form a Lie algebra with structure constants C ijk 1≤i,j,k≤r , i.e.

[v i , v j ] = r k=1 C ijk v k . (1)
The Maurer-Cartan forms1 ω = (ω 1 , . . . , ω r ) are the invariant one-forms that are dual to the chosen basis of invariant vector fields v = (v 1 , . . . , v r ) on G. The structure equations, dual to (1), are

dω k = - r 1≤i<j≤r C ijk ω i ∧ ω j . (2)
The Maurer-Cartan forms are at the heart of the Cartan's approach for equivalence of submanifolds. As observed in [START_REF] Weyl | Cartan on groups and differential geometry[END_REF][START_REF] Gary | Higher order contact of submanifolds of homogeneous spaces[END_REF] one can distinguish the fixed parameterization problem and the unparameterized problem. For the fixed parameterization problem Griffiths [START_REF] Griffiths | On cartan's method of lie groups as applied to uniqueness and existence questions in differential geometry[END_REF] proposed a reinterpretation of Cartan's moving frame method that works along the following lines. The monograph by Jensen [START_REF] Gary | Higher order contact of submanifolds of homogeneous spaces[END_REF] treats the unparameterized problem.

In Klein geometry one considers the action of G on a homogeneous space G/H, where H is a subgroup of G. Given two immersions f 1 , f 2 : X → G/H one wishes to determine when there exists λ ∈ G s.t.

f 1 (x) = λ • f 2 (x), for all x ∈ X .
A solution to the problem is based on the following result [START_REF] Griffiths | On cartan's method of lie groups as applied to uniqueness and existence questions in differential geometry[END_REF], [21, Theorem 1.6.10, Corolary 1.6.11].

Lemma 1.1 Let f1 , f2 : X → G be two smooth maps of a connected manifold X into the Lie group G. Then f1 (x) = λ • f2 (x), for a fixed λ ∈ G and for all x ∈ X , if and only if

f * 1 ω 1 = f * 2 ω 1 , . . . , f * 1 ω r = f * 2 ω r ,
where ω 1 , . . . , ω r are the Maurer-Cartan forms on G.

The solution to the equivalence problem relies then on finding an equivariant lift i.e. a process that associates to each immersion f :

X → G/H an immer- sion f : X → G s.t. λ • f = λ • f and π • f = f , for the natural projection π : G → G/H.
The lifts, the craft of which requires a good geometrical understanding, usually involve the jets of f . Differential invariants thus appear as coefficients when we write the invariant forms f * ω i in terms of an (invariant) coframe the for X . The lemma thus shows that those invariants separate orbits. In the fixed parametrization case, the forms on X are invariant, but intrinsic parametrisations are to be determined and used for generality.

In the first lines of this section we spoke of invariance on G without specifying if it was under left or right multiplication. In line with [START_REF] Fels | Moving coframes. II. Regularization and theoretical foundations[END_REF][START_REF] Hubert | Differential invariants of a Lie group action: syzygies on a generating set[END_REF] we consider invariance under right multiplication. The prolongation to higher order jets combined with the moving frame as defined in Section 2.3 methodologically provide the equivariant lift, the art residing only in choosing the most appropriate crosssection.

The approach is not restricted to homogeneous spaces. In line with [START_REF] Fels | Moving coframes. II. Regularization and theoretical foundations[END_REF] and in view of our original interest in differential equations, we consider an action on a space of dependent and independent variables. On a manifold of jets of high enough order, a local cross-section to the orbits determines a moving frame. The Maurer-Cartan invariants are the entries of the matrix relating the pullback of the Maurer-Cartan forms to a basis of contact invariant differential forms, which is given by the moving frame.

Normalized invariants

We go through the construction of normalized invariants based on the choice of a cross-section to the orbit of the prolonged action. This is in line with [START_REF] Fels | Moving coframes. II. Regularization and theoretical foundations[END_REF][START_REF] Hubert | Rational invariants of a group action. construction and rewriting[END_REF][START_REF] Hubert | Differential invariants of a Lie group action: syzygies on a generating set[END_REF] and we refer to those three papers for more details and examples.

Group action and prolongations

For a smooth manifold M we note F(M) the R-algebra of smooth functions on M. For a smooth map φ : M → N , we denote φ * : F(N ) → F(M) the pullback: (φ * f )(z) = f (φ(z)) for z ∈ M and f ∈ F(N ).

We consider a manifold X × U where X and U are covered by single coordinate systems, x = (x 1 , . . . , x m ) and u = (u 1 , . . . , u n ) respectively. The additional coordinate functions for the space of jets of order k, J k = J k (X , U), are the u i,α , where α = (α 1 , . . . , α m ) ∈ N m is such that |α| = m i=1 α i ≤ k, that correspond to the derivatives ∂ |α| ui ∂x α . We consider an action g of G on J 0 = X ×U and its prolongations to the jet spaces J k , g : G × J k → J k . We note V k = (V k 1 , . . . , V k r ) the infinitesimal generators associated to our choice of right invariant vector fields v = (v 1 , . . . , v r ) on G:

V k i (f )(z) = d dt t=0 f (g(e tv , z)), z ∈ J k , f ∈ F(J k ).
The expression of the prolongations, of the action or of the infinitesimal generators, can be derived explicitly from the expression of the action of g on J 0 , or of V 0 [START_REF] Olver | Applications of Lie Groups to Differential Equations[END_REF][START_REF] Olver | Equivalence, invariants and Symmetry[END_REF].

The maximal dimension r k of the orbits on J k can only increase as the action is prolonged to higher order jets. It can not go beyond the dimension of the group though. The stabilization order is the order at which the maximal dimension of the orbits becomes stationary. If the action on J 0 is locally effective on subsets, i.e. the global isotropy group is discrete, then, for s bigger than the stabilization order, the action on J s is locally free on an open subset of J s [41, Theorem 5.11]. We shall make this assumption of a locally effective action. The maximal dimension of the orbits in J s is then r, the dimension of the group. We have:

r 0 ≤ r 1 ≤ . . . ≤ r s-1 < r s = r s+1 = . . . = r. Definition 2.1 A differential invariant of order k is a function f of F(J k ) s.t. V k 1 (f ) = 0, . . . , V k r (f ) = 0.
The R-algebra of differential invariants of order k is noted F G (J k ) while F G (J) denotes the R-algebra of differential invariants of any order.

Local cross-section

For any k ∈ N, a local cross-section to the orbits of g in J k defines an invariantization and a set of normalized invariants on an open set of J k . Definition 2.2 An embedded submanifold P k of J k is a local cross-section to the orbits if there is a neighborhood U k of P k in J k such that -P k is of complementary dimension to the orbits in U k , -P k intersects at a unique point and transversally the connected part of the orbit of any z in U k .

The first condition implicitly implies that the action is semi-regular in U k , i.e. that all the orbits are of the same dimension. The results in this paper actually restrict to the open set U k . Though we shall keep the global notation, J k is understood to be restricted to this neighborhood of P k .

An embedded submanifold of codimension r k can be locally defined as the zero set of a map P : J k → R r k where the components (p 1 , . . . , p r k ) are independent functions of F(J k ). The condition for P to define a local cross section on J k is:

the rank of the r × r k matrix (V i (p j )) j=1..r k i=1..r is r k on P k . (3)
When G acts semi-regularly on J k there is thus a lot of freedom in choosing a cross-section. In particular we can always choose a coordinate cross-section [19, Theorem 5.6], i.e. the level set of an appropriate subset of the coordinate functions.

Let s be equal or greater than the stabilization order and let P s be a crosssection to the orbits in J s . Its pre-image P s+k in J s+k by the projection map π s+k s : J s+k → J s is a cross-section to the orbits in J s+k . In other words, if P s is the zero set in J s of p 1 , . . . , p r : J s → R then those same functions cut out a local cross-section for J s+k . Note though that the projection of P s on J k , for k < s, need not be a cross-section to the orbits in J k . Definition 2.3 A cross-section P k to the orbits in J k is of minimal order if, for any l ≤ k, its projection on J l is a local cross-section.

There is no harm in then assuming that a minimal order cross-section P k is defined by P = (p 1 , . . . , p r k ) where (p 1 , . . . , p r l ) defines a local cross-section on J l , for any l ≤ k. Definition 2.4 Let P k be a local cross-section to the orbits of the action g :

G × J k → J k . The invariantization ῑf of f ∈ F(J k ) is the function defined by ῑf (z) = f (z)
where z is the intersection of the orbit of z ∈ J k with P k . Definition 2.5 The normalized invariants of order k associated to a crosssection P k in J k are the invariantizations of the coordinate functions on J k . The set they form is denoted

I k = {ῑx 1 , . . . , ῑx m } ∪ {ῑu α | u ∈ U, |α| ≤ k}.
If we chose a coordinate cross-section, then some of those normalized invariants are constants. More generally, the normalized invariants are subject to the functional relationships implied by the choice of the equations of the crosssection. If the cross-section is defined by p 1 (x, u α ) = 0, . . . , p r k (x, u α ) = 0 then p 1 (ῑx, ῑu α ) = 0, . . . , p r k (ῑx, ῑu α ) = 0. This is a particular case of the following result.

Theorem 2.6 Assume ῑ is the invariantization associated to a cross-section P k in J k . For f ∈ F(J k ) ῑf is the unique differential invariant (of order k) whose restriction to P k is equal to the restriction of f to P k .

Corollary 2.7 For f ∈ F(J k ), ῑf (x, u α ) = f (ῑx, ῑu α ).
In particular, if f ∈ F G (J k ) then ῑf = f and f (x, u α ) = f (ῑx, ῑu α ). Therefore I k is a generating set for the differential invariants of order k in a functional sense. Note though that, unless P k is of minimal order, I k need not contain a set of generators for F G (J l ), for l < k.

Moving frame

As noted already by [START_REF] Griffiths | On cartan's method of lie groups as applied to uniqueness and existence questions in differential geometry[END_REF][START_REF] Green | The moving frame, differential invariants and rigidity theorems for curves in homogeneous spaces[END_REF][START_REF] Gary | Higher order contact of submanifolds of homogeneous spaces[END_REF][START_REF] Ivey | Cartan for beginners: differential geometry via moving frames and exterior differential systems[END_REF], the geometric idea of classical moving frames, like the Frenet frame for space curves in Euclidean geometry, can be understood as maps to the group. Accordingly Fels and Olver [START_REF] Fels | Moving coframes. II. Regularization and theoretical foundations[END_REF] defined a moving frames as an equivariant map ρ : J s → G, for s big enough, as they gave a new interpretation of Cartan's repère mobile [START_REF] Cartan | La méthode du repère mobile, la théorie des groupes continus, et les espaces généralisés[END_REF][START_REF] Cartan | La théorie des groupes finis et continus et la géométrie différentielle traitées par la méthode du repère mobile[END_REF] .

Theorem 2.8 A Lie group G acts locally freely on J s if and only if every point of J s has an open neighborhood U s where there exists a locally equivariant map ρ :

U s → G in the sense that, for any z ∈ U s , ρ(λ • z) = ρ(z) • λ -1 for λ ∈ G sufficiently close to the identity.
As before, we shall use the global notation J s but we shall keep in mind that we are restricting to U s .

If the action g of G is locally free on J s then a local cross-section P s to the orbits defines a moving frame. Indeed the equation g(ρ(z), z) ∈ P s for z ∈ J s and ρ(z) = e, ∀z ∈ P s uniquely defines a smooth map ρ : J s → G in a sufficiently small neighborhood of any point of the cross-section. This map is seen to be equivariant. If P is the zero set of the map P = (p 1 , . . . , p r ) then p 1 (g(ρ, z)) = 0, . . . , p r (g(ρ, z)) = 0 are implicit equations for the moving frame. If we can solve those, the moving frame ρ provides an explicit construction for the invariantization process. To spell it out let us introduce the following maps.

σ : J s+k → G × J s+k z → (ρ(z), z) and π = g • σ : J s+k → J s+k z → g(ρ(z), z) (4)
Proposition 2.9 For f ∈ F(J s+k ), ῑf = π * f, that is ῑf (z) = f (g(ρ(z), z)) for all z ∈ J s+k .
This is a restatement of [START_REF] Hubert | Smooth and algebraic invariants of a group action. Local and global constructions[END_REF]Proposition 1.16]. An invariantization of forms is defined in [START_REF] Kogan | Invariant Euler-Lagrange equations and the invariant variational bicomplex[END_REF]. Besides the map σ and g it involves a projection on the jet differentials.

Invariant derivations and forms

The moving frame defined by a cross-section allows to construct a frame of invariant derivations, and, equivalently, their dual contact invariant coframe.

The coordinates of the horizontal part of pullback of the Maurer-Cartan forms in this coframe are theMaurer-Cartan invariants. They are the entries of the m×r matrix K that can be explicitely written in terms of the normalized invariants with the only knowledge of the infinitesimal generators and the equations of the cross-section. The Maurer-Cartan invariants arise in then in the explicit formula for the invariant derivation of normalized invariants. The material of this section draws in an essential way on [START_REF] Fels | Moving coframes. II. Regularization and theoretical foundations[END_REF][START_REF] Kogan | Invariant Euler-Lagrange equations and the invariant variational bicomplex[END_REF][START_REF] Hubert | Differential invariants of a Lie group action: syzygies on a generating set[END_REF] where examples can be found.

The vector D = (D 1 , . . . , D m ) T of total derivations

D i = ∂ ∂x i + u,α u α+ i ∂ ∂u α
is dual to the horizontal one forms dx = (dx 1 , . . . , dx m ) T . Let D = (D 1 , . . . , D m ) T be a vector of total derivations defined by D = A -1 D where A ∈ F(J k+1 ) is an invertible matrix. The dual horizontal forms x = (x 1 , . . . , x m ) T are then given by x = A T dx.

For λ ∈ G, let g λ : J k → J k be the map defined by g λ (z) = g(λ, z). Definition 5.25]. Dual to contactinvariant forms, invariant derivations are total derivations that commute with the infinitesimal generators of the group action. It maps differential invariants of order k to differential invariants of order k + 1, for k large enough.

A differential form ν on J k is invariant if g * λ (ν| g λ (z) ) = ν| z , for all λ ∈ G. It is contact invariant if g * λ (ν| g λ (z) ) -ν| z is a contact form [41,
For instance, if f is a differential invariant then df is an invariant differential form while its projection on the horizontal coframe, d H f , is contact invariant. A frame of invariant derivations can thus be constructed from sufficiently many independent differential invariants. Additionally, if ρ : J s → G is a moving frame, then the pullback of the Maurer-Cartan forms, ρ * ω i , 1 ≤ i ≤ r are invariant forms.

The construction of a contact-invariant coframe x = (x[1], . . . , x[m]) proposed by [START_REF] Fels | Moving coframes. II. Regularization and theoretical foundations[END_REF] is based on a moving frame. Let P = (p 1 , . . . , p d ) defines a cross-section P to the orbits in J s , where s is greater than the stabilization order. Consider ρ : J s → G the associated moving frame and ῑ : F(J) → F G (J) the associated invariantization. Define σ : J s → G × J s by σ(z) = (ρ(z), z). Theorem 3.1 The vector of derivations D = (σ * A) -1 D, where A is the m × m matrix (D i (g * x j )) ij , is a vector of invariant derivations. Equivalently, x = (σ * A) T dx is a vector of contact invariant forms.

Though those invariant derivations do not commute in general, their formidable benefit is an explicit formula for the derivation of a normalized invariants. This formula involves the Maurer-Cartan invariants.

We denote by D(P ) the m × r matrix (D i (p j )) i,j with entries in F(J s+1 ) while V(P ) is the r × r matrix (V i (p j )) i,j with entries in F(J s ). As P is transverse to the orbits in J s , the matrix V(P ) has non zero determinant along P and therefore in a neighborhood of each of its points. Theorem 3.2 Consider D = (D 1 , . . . , D m ) T , and x = (x 1 , . . . , x m ) T , the vector of invariant derivations, and their dual contact-invariant forms, constructed in Theorem 3.1. Let K be the m × r matrix obtained by invariantizing the entries of D(P ) V(P ) -1 . Then

ρ * ω = -K T x modulo contact forms and D(ῑf ) = ῑ(Df ) -K ῑ(V(f )).
As they relate the horizontal part of the pullback of the Maurer-Cartan forms ρ * ω to the contact invariant coframe x of Theorem 3.1, the entries of the matrix K are the Maurer-Cartan invariants

The statement can be deduced from [7, Section 13]. We extend the proof of [START_REF] Hubert | Differential invariants of a Lie group action: syzygies on a generating set[END_REF] that takes the dual approach.

proof: From the definition of σ : z → ( ρ(z), z ) and the chain rule we have

D(ῑf )(z) = D(σ * g * f )(z) = D(g * f )(ρ(z), z) + (ρ * D)(g * f )(ρ(z), z). (5)
Consider the matrix A of Theorem 3.1. The vector of derivations D = A -1 D has the following property: Dj (g * f ) = g * (D j f ) for all f ∈ F(J). It is indeed used to compute the prolongations [41, Chapter 4], [16, Section 1.2]. We have

D(g * f )(ρ(z), z) = (σ * D(g * f ))(z) = σ * g * (Df )(z) = ῑ(Df )(z) and (5) becomes D(ῑf )(z) = ῑ(Df )(z) + σ * (ρ * D)(g * f )(z). (6)
Let v = (v 1 , . . . , vd ) be the infinitesimal generators of the action of G on G by the right multiplication. Besides being left invariant, they form a basis for the derivations on G. There is thus a matrix K with entries in

F(G × J s ) s.t. ρ * D = K v.
This latter equation shows that σ * KV(g * f ) = ῑ(Df ) -D(ῑf ) is a differential invariant. As such it is equal to its invariantization and thus

σ * KV(g * f ) = ῑ(σ * K) ῑ(σ * V(g * f )).
For all z ∈ P , ρ(z) = e and therefore σ * V(g * f ) and V(f ) agree on P : for all

z ∈ P , σ * V(g * f ) (z) = V(g * f ) (e, z) = V(f )(z) by [16, Proposition 1.1]. It follows that ῑ(σ * V(g * f )) = ῑ(V(f )) so that (7) becomes D(ῑf )(z) = ῑ(Df )(z) -ῑ(σ * K) ῑ(V(f )). ( 8
)
To find the matrix K = ῑ(σ * K) we use the fact that ῑp i = 0 for all 1 ≤ i ≤ r.

Applying D and ( 8) to this equality we obtain: ῑ(Dp i ) = K ῑ(V(p i )) so that ῑ(D(P )) = K ῑ(V(P )). The transversality of P imposes that V(P ) is invertible along P , and thus so is ῑ(V(P )).

We thus have proved that D(ῑf

) = ῑ(Df ) -K ῑ(V(f )) where K = ῑ(σ * K) = ῑ(D(P )V (P ) -1 ).
Since ρ * ω is a vector of invariant forms while x is a vector of contact invariant forms whose entries span the horizontal coframe there is a matrix A with differential invariants as entries such that ρ * ω = Ax modulo contact forms . We prove that A = -K T .

Since x and D are dual we have A ai (z) = ρ * ω a , D i = ω a , ρ * D i which can be written in matrix form as A T = ω T (ρ * D) and thus, taking forms and vector fields where they should be taken, A T = ω| T ρ(z) K(ρ(z), z)v| ρ(z) . In particular for z ∈ P we have A(z) T = ω T e K(e, z)v| e . As v| e = -v| e and v is dual to ω we have A(z) T = -K(e, z) for all z ∈ P . Therefore A = -ῑ(σ * K) T and we proved that σ * K = K. 2

The commutators of the invariant derivations is obtained explicitly by deriving a recurrence formula for forms in [7, Section 13]. It can also be derived directly from Theorem 3.2 through the use of formal invariant derivations [START_REF] Hubert | Algebra of differential invariants[END_REF].

Proposition 3.3 For all 1 ≤ i, j ≤ m, D i , D j = m k=1 Λ ijk D k where Λ ijk = d c=1 K ic ῑ(D j (ξ ck )) -K jc ῑ(D i (ξ ck )) ∈ F G (J s+1 ), K = ῑ D(P ) V (P ) -1 , and ξ ck = V c (x k ).

Generation

The formula for the derivation of invariantized function of previous section is the key to the generation property of three different sets of differential invariants, among which the Maurer-Cartan invariants.

If we specialize Theorem 3.2 for f = x i of f = u α we obtain the so called recurrence formulae [7, Section 13]:

D i (ῑx j ) = δ ij - r a=1 K ia ῑ (V(x j )) 1 ≤ i, j, ≤ m D i (ῑu α ) = ῑu α+ i + r a=1 K ia ῑ (V(u α )) , ∀α ∈ N m , 1 ≤ i ≤ m
As the entries of K belong to F G (J s+1 ), it immediately follows from those recurrence formulae that the normalized invariants of order s + 1 and less form a generating set of differential invariants. This set has cardinality m+n m+s+1 s+1 . If we restrict to cross-sections of minimal order we can find a generating set of cardinality bounded by m r + d 0 where d 0 is the codimension of the orbits in J 0 . In the case of coordinate cross-section it is a subset of I s+1 and the result appeared in [START_REF] Olver | Generating differential invariants[END_REF]. The general case appeared in [START_REF] Hubert | Differential invariants of a Lie group action: syzygies on a generating set[END_REF]Theorem 4.2]. Theorem 4.1 If P = (p 1 , . . . , p r ) defines a cross-section for the action of g on J such that P k = (p 1 , . . . , p r k ) defines a cross-section for the action of g on J k , for all k, then the edge invariants E = {ῑ(D i (p j )) | 1 ≤ i ≤ m, 1 ≤ j ≤ r} together with I 0 form a generating set of differential invariants.

As discussed in [16, Section 4 and 5] the rewriting in terms of the normalized invariants of I s+1 and their derivatives is effective and relies on a simple inductive process. The proof of [START_REF] Hubert | Differential invariants of a Lie group action: syzygies on a generating set[END_REF]Theorem 4.2] shows that the rewriting in terms of E can be performed with linear algebra operations.

A new observation is that, without the assumption of minimal order on the cross-section, we can always find a generating set of cardinality bounded by m r + d 0 , where d 0 is the dimension of the projection of the cross-section on J 0 . Indeed the Maurer-Cartan invariants form a generating set. The proof of the following theorem provides the inductive process by which the rewriting is made effective. Theorem 4.2 Take s to be equal to the stabilization order or greater. Let P s be a cross-section on J s defined by the map P = (p 1 , . . . , p r ). The union of 0 th order normalized invariants, I 0 , and

K = {K ia | 1 ≤ i ≤ m, 1 ≤ a ≤ r}
, the entries of the matrix K = ῑ(D(P )V (P ) -1 ), form a generating set of differential invariants.

proof: From Theorem 3.2, ῑ(Du α ) = D(ῑu α ) + K ῑ(V(u α )) for any α ∈ N m and u ∈ U. V(u α ) is a vector of function of F(J |α| ). Therefore the invariantization of all the coordinate functions of order k + 1 can be written in terms of the differential invariants of order k, and their derivatives, together with the http://hal.inria.fr/inria-00194528 differential invariants coming as entries of the matrix K. An inductive argument brings the conclusion. 2

The case of transitive action on J 0 has a special importance. In this case the cross-section on J 0 can be chosen as a single point and the zero-th order normalized invariants are then constant. We obtain then the striking result that the Maurer-Cartan invariants form a generating set.

Corollary 4.3 Assume the action is transitive on J 0 and choose a cross-section the projection of which provides a cross-section on J 0 . The Maurer-Cartan invariants form then a generating set of differential invariants.

Syzygies

The Maurer-Cartan invariants, as well as the edge invariants, are differential invariants of order s + 1, where s is the order of the chosen moving frame. They need not to be functionally independent. Yet we know all the functional relationships on the normalized invariants: they are given by the functions the invariantization of which is zero. They are the linear combination of the r functions cutting out the cross-section [START_REF] Hubert | Differential invariants of a Lie group action: syzygies on a generating set[END_REF]Proposition 2.12]. If the coefficients of infinitesimal operators are rational functions while the cross-section is cut out by polynomial functions, we can algorithmically compute the functional relationships among the Maurer-Cartan invariants by algebraic elimination [START_REF] Becker | Gröbner Bases -A Computational Approach to Commutative Algebra[END_REF][START_REF] Cox | Ideals, Varieties, and Algorithms[END_REF][START_REF] Kreuzer | Computational commutative algebra[END_REF][START_REF] Greuel | A Singular introduction to commutative algebra[END_REF].

Similarly we know a complete set of differential syzygies for I s+1 [START_REF] Hubert | Differential invariants of a Lie group action: syzygies on a generating set[END_REF]Theorem 5.13]. By differential elimination [START_REF] Hubert | Notes on triangular sets and triangulation-decomposition algorithms II: Differential systems[END_REF][START_REF] Hubert | Differential algebra for derivations with nontrivial commutation rules[END_REF][START_REF] Boulier | diffalg: description, help pages and examples of use[END_REF][START_REF] Hubert | diffalg: extension to non commuting derivations[END_REF] we can determine a complete set of differential syzygies for the edge invariants E and for the Maurer-Cartan invariants K. Yet the Maurer-Cartan invariants carry the structure equations that provides their syzygies.

The Maurer-Cartan forms satisfy the structure equations provided by the structure constants C abc of the Lie algebra of right invariant vector fields on G.

dω c = - a<b C abc ω a ∧ ω b where [v a , v b ] = C abc v c (9)
It follows that their pull-backs ρ * ω satisfy the same closed exterior differential system. Let Θ be the contact ideal and µ a , 1 ≤ a ≤ r, be the horizontal part of ρ * ω a , i.e. ρ * ω = µ a mod Θ. It satisfy thus:

dµ c mod Θ = - a<b C abc µ a ∧ µ b .
Rewriting this equation in terms of x thus provides differential relationships satisfied by the Maurer-Cartan invariants. We give them explicitly.

Proposition 5.1 The Maurer-Cartan invariants are subject to the relationships

∆ cij : D j (K ic ) -D i (K jc ) + 1≤a<b≤r C abc (K ia K jb -K ja K ib ) + m k=1 Λ ijk K kc = 0
where C abc are the structure constants ( 9) and Λ ijk are the coefficients of the commutations rules for the invariant derivations (Theorem 3.3).

proof: From Theorem 3.2 we know that

µ c ≡ -k K kc x k . Since dx k = -i<j Λ ijk x i ∧ x j mod Θ we have dµ c mod Θ = i<j D j (K ic ) -D i (K jc ) + m k=1 Λ ijk K kc x i ∧ x j while µ a ∧ µ b = i<j (K ia K jb -K ib K ja ) x i ∧ x j
Equating the coefficients of

x i ∧ x j in dµ c = -C abc µ a ∧ µ b we obtain ∆ cji . 2
In the case of curves, when there is only one independent variable, Proposition 5.1 predicts no syzygies. In Section 7.1 we exhibit a simple case where the non minimality of the cross-section implies a non trivial syzygies on the Maurer-Cartan invariants. In Section 7.2 we exhibit an example involving two independent variables where Proposition 5.1 does not provide a complete set of syzygies when the cross-section is not chosen of minimal order. We conjecture that in the case of an action transitive on J 0 and with the choice of a cross-section of minimal order, the syzygies of Proposition 5.1 form a complete set, i.e. any other differential relationships among the monotone derivatives of the Maurer-Cartan invariants belongs to the differential ideal they generate.

The conjecture has been verified on classical examples by applying differential elimination [START_REF] Hubert | Notes on triangular sets and triangulation-decomposition algorithms II: Differential systems[END_REF][START_REF] Boulier | diffalg: description, help pages and examples of use[END_REF][START_REF] Hubert | Differential algebra for derivations with nontrivial commutation rules[END_REF][START_REF] Hubert | diffalg: extension to non commuting derivations[END_REF] on the complete set of differential syzygies for the normalized invariants I s+1 given in [START_REF] Hubert | Differential invariants of a Lie group action: syzygies on a generating set[END_REF].

Matrix group

Because of the explicit form that can be achieved for matrix groups, the Maurer-Cartan form is often defined to be the one-form Ω = r a=1 ω a ⊗ v a | e with value in the Lie algebra of G. The structure equation can then be written compactly dΩ = -1 2 [Ω, Ω] (see for instance [START_REF] Ivey | Cartan for beginners: differential geometry via moving frames and exterior differential systems[END_REF]). In this section we specialize the results about Maurer-Cartan invariants to the case where the group is known by a representation. It is in particular intensely used in the evolution of curves in Klein geometries and their relationships to integrable systems (see [START_REF] Beffa | The theory of differential invariants and KdV Hamiltonian evolutions[END_REF][START_REF] Beffa | Poisson brackets associated to invariant evolutions of Riemannian curves[END_REF][START_REF] Beffa | Poisson geometry of differential invariants of curves in some nonsemisimple homogeneous spaces[END_REF][START_REF] Mansfield | Evolution of curvature invariants and lifting integrability[END_REF][START_REF] Beffa | On completely integrable geometric evolutions of curves of lagrangian planes[END_REF][START_REF] Beffa | Projective-type differential invariants and geometric curve evolutions of kdv-type in flat homogeneous manifolds[END_REF] and references therein). As a way of making the connection, we set as goal to show how the results we have obtained are linked with the results of the first part of [START_REF] Mansfield | Evolution of curvature invariants and lifting integrability[END_REF]. In Section 7.3 we discuss a classical example in both the general formalism and in the matricial setting.

Let τ : G → Gl(d) be a representation of the Lie group G. The Lie algebra of τ (G) can be understood as a subspace of the d × d matrices. Let (a 1 , . . . , a r ) be the d × d matrices that are images by τ * at identity of the right invariant vector fields (v 1 , . . . , v r ) on G. The Maurer-Cartan form Ω on τ (G) can then be interpreted as a d × d matrix with one-form as entries. If ω 1 , . . . , ω r are the right invariant one-forms that are dual to v 1 , . . . , v r then Ω = r a=1 ω a a a .

One shows that2 Ω = (dτ ) (τ ) -1 and the structure equation is given by d Ω = -Ω ∧ Ω.

Consider an action g : G × J 0 → J 0 effective on subsets, and its prolongations to the higher order jet spaces J k , for any k ∈ N. Assume that s is greater than the stabilization order and consider a cross-section P to the orbits on J s cut out by P = (p 1 , . . . , p r ), p i ∈ F(J s ). Let ρ : J s → G be the associated moving frame and ρ = τ • ρ : J s → Gl(d).

The matrices

Q i = D i (ρ) ρ-1 , 1 ≤ i ≤ m,
were defined in [START_REF] Mansfield | Evolution of curvature invariants and lifting integrability[END_REF] as the curvature matrices extending the concepts of [START_REF] Hitchin | of Oxford Graduate Texts in Mathematics[END_REF]. In the case of curves (m = 1) it is the Serret-Frenet matrix [START_REF] Beffa | Differential invariants for parametrized projective surfaces[END_REF][START_REF] Beffa | The theory of differential invariants and KdV Hamiltonian evolutions[END_REF][START_REF] Beffa | Relative and absolute differential invariants for conformal curves[END_REF][START_REF] Beffa | Projective-type differential invariants and geometric curve evolutions of kdv-type in flat homogeneous manifolds[END_REF]. Proposition 6.1 The curvature matrices satisfy Q i = r a=1 K ia a a where K is the matrix of Theorem 3.2 i.e. K = ῑ D(P )V(P ) -1 .

The result was proved in [START_REF] Mansfield | Evolution of curvature invariants and lifting integrability[END_REF] by calculus and we show how it can be recovered by Theorem 3.2.

proof:

On one hand ρ * ω a = -m i=1 K ia x i mod Θ, according to Theorem 3.2, and therefore

ρ * Ω = r a=1 ρ * ω a a a = r a=1 m i=1 K ia x i a a mod Θ.
On the other hand, D and x are dual so that dρ =

m i=1 D i (ρ) x i mod Θ and thus ρ * Ω = (dρ) ρ-1 = m i=1 Q i x i mod Θ.
We thus obtain the announced result. 2 Corollary 4.3 of Theorem 4.2 furthermore informs us that the invariants in the curvature matrices Q form a generating set when the action is transitive on J 0 .

To conclude let us note that by substituting ρ * Ω = Q i x i in the structure equation d(ρ * Ω) = -ρ * Ω ∧ ρ * Ω we obtain the syzygies on the curvature matrices

Q i : D j (Q i ) -D i (Q j ) = Λ ijk Q k + Q j Q i -Q i Q j
given in [31, proposition 9] to generalize the zero-curvature equation [START_REF] Hitchin | of Oxford Graduate Texts in Mathematics[END_REF] that applies only when the derivations D commute. This is thus a matricial form of the syzygies of Proposition 5.1.

Examples

We first give two examples where computations are easily performed, one for curves and and one for surfaces. Those examples are variations of the running examples in [START_REF] Hubert | Differential invariants of a Lie group action: syzygies on a generating set[END_REF]. We then treat the two very classical geometries, curves and surfaces in Euclidean 3-space, in order to illustrate the general theory of this paper. With the knowledge of the infinitesimal generators of the action and a choice of cross-section only, we can characterize a set of generators and compute their syzygies. While non trivial applications are presented in [START_REF] Hubert | Differential invariants of conformal and projective surfaces[END_REF], this section is only meant for understanding.

Curves

This example illustrates first the material in Section 2 and 4 for a minimal order cross-section. We then move to a cross-section that is not of minimal order. The Maurer-Cartan invariants still form a generating set, but there is a differential relationship among them that is not predicted by Proposition 5.1.

Group action

We consider the group

G = R * R 2 with multiplication (λ 1 , λ 2 , λ 3 )•(µ 1 , µ 2 , µ 3 ) = (λ 1 µ 1 , λ 2 + λ 1 µ 2 , λ 3 + µ 3 ).
A basis of right invariant vector fields is given by [START_REF] Olver | Equivalence, invariants and Symmetry[END_REF]Example 2.46]

v 1 = λ 1 ∂ ∂λ 1 + λ 2 ∂ ∂λ 2 , v 2 = ∂ ∂λ 2 , v 3 = ∂ ∂λ 3
so that the dual Maurer-Cartan forms are:

ω 1 = 1 λ 1 dλ 1 , ω 2 = - λ 2 λ 1 dλ 1 + dλ 2 , ω 3 = dλ 3
We consider the action g of G on X 1 × U 1 given by:

g * x = λ 1 x + λ 2 , g * u = u + λ 3 .
http://hal.inria.fr/inria-00194528

It extends to derivatives by g

* u k = u k λ k 1
, for k > 0. The infinitesimal generators of this prolonged action are given by:

V 1 = x ∂ ∂x -k u k ∂ ∂u k , V 2 = ∂ ∂x , V 3 = ∂ ∂u 0 .

Minimal order cross-section

Let us consider first the cross-section defined by P = (x, u 0 , u 1 -1). The Maurer-Cartan matrix is

K = ῑ D(P ) V (P ) -1 = ( -ῑu 2 1 1 ).
By Theorem 4.2 (or Theorem 4.1) we deduce that {ῑu 2 } is a generating set of differential invariants. By Theorem 3.2 we know that:

D(ῑu i ) = ῑu i+1 + ῑu 2 ῑ(V 1 (u i )) -ῑ(V 2 (u i )) -ῑ(V 3 (u i ))
so that, for i > 0, D(ῑu i ) = ῑu i+1 -i ῑu 2 ῑu i .

We can verify the recurrence formulae above with the explicit form of the moving frame associated to P = (x, u 0 , u 1 -1). It is indeed given by

ρ * λ 1 = u 1 , ρ * λ 2 = -x u 1 , ρ * λ 3 = -u 0
so that the normalized invariants and invariant derivation are:

ῑu i = u i u i 1 and D = 1 u 1 D.

Non minimal order cross-section

Let us consider now the cross-section defined by P = (x, u 0 , u 2 -1). The Maurer-Cartan matrix is

K = ῑ D(P ) V (P ) -1 = ( ψ 1 φ ) where ψ = - 1 2 ῑu 3 , φ = ῑu 1 .
By Theorem 3.2 we know that:

D(ῑu i ) = ῑu i+1 -ψ ῑ(V 1 (u i )) -ῑ(V 2 (u i )) -φ ῑ(V 3 (u i )) so that, for i > 0, D(ῑu i ) = ῑu i+1 -i ψ ῑu i (10)
By Theorem 4.2 we deduce that {φ, ψ} is a generating set of differential invariants. Applying [START_REF] Griffiths | On cartan's method of lie groups as applied to uniqueness and existence questions in differential geometry[END_REF] for i = 1 we obtain D(ῑu 1 ) = 1 + ψ ῑu 1 so that φ ψ + D(φ) -1 = 0.

On one hand this proves that {φ} is a generating set. On the other hand this latter equation is a syzygie that is not predicted by Proposition 5.1.

We can verify the recurrence formula above as the explicit form of the moving frame associated to P = (x, u 0 , u 2 -1) is given by

ρ * λ 1 = √ u 2 , ρ * λ 2 = -x √ u 2 , ρ * λ 3 = -u 0 .
The normalized invariants and invariant derivation are thus

ῑu i = u i u i/2 2 and D = 1 √ u 2 D.

Surfaces

We move now to the case of more than one independent variable. This example illustrates first the material discussed in Section 2, 4, and 5 for a minimal order cross-section. We then move to a cross-section that is not of minimal order.

Group action

We consider the group G = R * R 3 with action on X 2 × U 1 given by

g * x = λ 1 x + λ 2 , g * y = λ 1 y + λ 3 , g * u = u + λ 4 .
As mentionned earlier we only need the infinitesimal generators of the action as input data. Their prolongations to J are:

V 1 = x ∂ ∂x + y ∂ ∂y - i,j≥0 (i + j) u ij ∂ ∂u ij , V 2 = ∂ ∂x , V 3 = ∂ ∂y , V 4 = ∂ ∂u .

Minimal order cross-section

The cross-section cut out by P = (x, y, u 00 , u 10 -1) is of minimal order. The associated Maurer-Cartan matrix is

K = τ 1 0 1 σ 0 1 ψ where ψ = ῑu 01 , τ = -ῑu 20 , σ = -ῑu 11 .
Theorem 3.2 implies that for i + j > 0

D 1 (ῑu ij ) = ῑu i+1,j + (i + j) τ ῑu ij , D 2 (ῑu ij ) = ῑu i,j+1 + (i + j) σ ῑu ij while, by Theorem 3.3, [D 1 , D 2 ] = σ D 1 -τ D 2
Theorem 4.2 implies that {ψ, τ, σ} is a generating set of differential invariants. Proposition 5.1 indicates furthermore that

D 1 (τ ) -D 2 (σ) = 0, D 1 (ψ) = τ ψ -σ.
http://hal.inria.fr/inria-00194528

From the second equation we deduce that σ = τ ψ -D 1 (ψ) so that {ψ, τ } is already a generating set. It is subject to a differential syzygy of order 2 that is obtained by substituting τ in the first equation.

Non minimal cross-section

The cross-section cut out by P = (x, y, u 00 , u 20 -1) is of minimal order. The associated Maurer-Cartan matrix is

K = τ 1 0 φ σ 0 1 ψ where φ = ῑu 10 , ψ = ῑu 01 , τ = - 1 2 ῑu 30 , σ = - 1 2 ῑu 21 .
Theorem 4.2 implies that {φ, ψ, τ, σ} is a generating set of differential invariants and Proposition 5.1 indicates

D 1 (τ ) -D 2 (σ) = 0, D 1 (ψ) -D 2 (φ) = σ φ -τ ψ.
From the second equation we deduce that {φ, ψ, τ } can be taken as generating set.

Theorem 3.2 implies that for i + j > 0

D 1 (ῑu ij ) = ῑu i+1,j + (i + j) τ ῑu ij , D 2 (ῑu ij ) = ῑu i,j+1 + (i + j) σ ῑu ij while, by Theorem 3.3, [D 1 , D 2 ] = τ D 1 -σD 2 .
Applying the recurrence formula for i = 1, j = 0 we obtain D 1 (ῑu 10 ) = 1 + τῑu 10 i.e. D 1 (φ) = 1 + τ φ from which we can draw τ in terms of φ. Therefore {φ, ψ} is already a generating set subject to a differential syzygy of order 2.

Curves in Euclidean geometry

Group action

We consider the classical action of SE(3) on space curves. We have J 0 = X 1 ×U 2 with coordinate (x, u, v). The infinitesimal generators of the action are:

V 0 1 = ∂ ∂x , V 0 2 = ∂ ∂u , V 0 3 = ∂ ∂v V 0 4 = v ∂ ∂u -u ∂ ∂v , V 0 5 = x ∂ ∂u -u ∂ ∂x , V 0 6 = x ∂ ∂v -v ∂ ∂x
so that their prolongations are given by

V 1 = ∂ ∂x , V 2 = ∂ ∂u , V 3 = ∂ ∂v , V 4 = k v k ∂ ∂u k -u k ∂ ∂v k , V 5 = -u 0 D x + k D k (x -u 0 u 1 ) ∂ ∂u k - k D k (u 0 v 1 ) ∂ ∂v k , V 6 = -v 0 D - k D k (v 0 u 1 ) ∂ ∂u k + k D k (x -v 0 v 1 ) ∂ ∂v k
The action is transitive on J 1 and becomes locally free on J 2 with generic orbits of codimension 1.

Generation with non mininimal cross-section

We consider the non-minimal coordinate cross-section proved to fail Theorem 4.1 in [42, Section 6]:

P = (x, u 0 , v 0 , v 1 , v 2 , v 3 -1)
We have With the complete set of syzygies on I s+1 [16, Theorem 5.13], we find by differential elimination [START_REF] Hubert | Differential algebra for derivations with nontrivial commutation rules[END_REF][START_REF] Hubert | diffalg: extension to non commuting derivations[END_REF] that {ῑu 1 , ῑu 2 , ῑw} are algebraically dependent. More precisely we have:

K = 1 ῑu 1 0 - 1 ῑu 2 1 3 ῑ(u 2 v 4 -u 3 ) ῑ(u 1 u 2 ) - ῑ(u 1 ) ῑ(u 2
ῑu 2 2 -ῑu 1 ῑu 2 - 1 3 ῑw (1 + ῑu 2 1 ) ῑu 1 = 0
Therefore {ῑu 1 , ῑu 2 } form a generating set of differential invariants. Indeed, given the analytic expression for the curvature and the torsion [START_REF] Guggenheimer | Differential geometry[END_REF] we can rewrite them in terms of the normalized invariants by a simple substitution (Theorem 2.7). We obtain:

κ = ῑu 2 2 (1 + ῑu 2 1 ) 3 , τ = 1 ῑu 2 (1 + ῑu 2 1 )
. http://hal.inria.fr/inria-00194528

Minimal order cross-section and matrix representation

The cross-section cut out by P = (x, u 0 , v 0 , u 1 , v 1 , v 2 ) is of minimal order. By Theorem 4.1 we know that {ῑv 2 , ῑu 3 } is a generating set of differential invariants. On one hand, applying the replacement property of normalized invariants (Corollary 2.7) to the explicit expression of the curvature and the torsion we obtain: κ = ῑu 2 , τ = ῑv 3 ῑu 2 .

On the other hand the Maurer-Cartan matrix can be made explicit in terms of the normalized invariants, so that we can now recognize its expression in terms of the curvature and the torsion:

K = 1 0 0 - ῑv 3 ῑu 2 ῑu 2 0 = 1 0 0 -τ κ 0
A representation of SE( 3) is given by the matrix group

1 0 λ R λ ∈ R 3 , R ∈ SO(3) .
The Lie algebra of this matrix group is then spanned by

Surfaces in Euclidean geometry

Group action

We choose coordinate functions (x, y, u) for R 2 × R, i.e. we consider x, y as the independent variables and u as the dependent variable. The infinitesimal generators of the classical action of the Euclidean group E(3) on R 3 are:

V 1 = ∂ ∂x , V 2 = ∂ ∂y , V 3 = ∂ ∂u , V 4 = x ∂ ∂u -u ∂ ∂x , V 5 = y ∂ ∂u -u ∂ ∂y , V 6 = x ∂ ∂y -y ∂ ∂x .
The non zero structure constants are given by the following commutators of the infinitesimal generators:

[V 1 , V 4 ] = V 3 , [V 1 , V 6 ] = V 2 , [V 2 , V 5 ] = V 3 , [V 2 , V 6 ] = -V 1 , [V 3 , V 4 ] = -V 1 , [V 3 , V 5 ] = -V 2 , [V 4 , V 5 ] = -V 6 , [V 4 , V 6 ] = V 5 , [V 5 , V 6 ] = -V 4 .

Minimal order cross-section and syzygies

Let us choose the classical cross-section defined by P = (x, y, u, u 10 , u 01 , u 11 ). The Maurer-Cartan matrix is then http://hal.inria.fr/inria-00194528

Our generating invariants {κ, τ } are thus the principal curvatures. Substituting [START_REF] Guggenheimer | Differential geometry[END_REF] in ∆ 612 , we retrieve the Codazzi equation as their syzygy:

D 2 D 2 (κ) κ -τ -D 2 D 2 (τ ) κ -τ = D 2 (κ) κ -τ 2 + D 2 (τ ) κ -τ 2 + κτ.

  a i and V i are associated to the same right invariant vector field v i on G. We can determine the curvature matrix Q from Proposition 6.1, and recognize there the equations for the Serret-Frenet frame [21, Chapter 1]:Q = a 1 + κ a 5 -τ a 4 =

2 . 11 )= 1 2 ( 1 + u 2 01 )u 20 -2 10 u 01 u 11 +

 2112122011 [START_REF] Hubert | Differential invariants of conformal and projective surfaces[END_REF] , τ = ῑu 02 , φ = ῑu 21 ῑu 20 -ῑu 02, and ψ = ῑu 12 ῑu 20 -ῑu 02 .By Theorem 3.3 we have[D 2 , D 1 ] = φ D 1 + ψ DThe non zero syzygies of Proposition 5.1 are:∆ 412 : D 2 (κ) -φ (κ -τ ) = 0 ∆ 512 : -D 1 (τ ) -ψ (τ -κ) = 0 ∆ 612 : D 2 (φ) -D 1 (ψ) -κ τ -φ 2 -ψ 2 = 0The first two syzygies imply thatφ = D 2 (κ) κ -τ , ψ = D 1 (τ ) κ -τ . (It follows that {κ, τ } form a generating set. Indeed, from their analyic expression[START_REF] Guggenheimer | Differential geometry[END_REF],[21, Section 1.1] we can write the Gauss and mean curvatures in terms of the normalized invariants, thanks to Theorem 2.7, and eventually in terms of the Maurer-Cartan invariants by Theorem 3.2 and 4.2: σ = u 20 u 02 -u 11 (1 + u 2 10 + u 2 01 ) 2 = ῑu 20 ῑu 02 = κ τ, π

In Section 6 we link this definition to the one used[10,

[START_REF] Ivey | Cartan for beginners: differential geometry via moving frames and exterior differential systems[END_REF] http://hal.inria.fr/inria-00194528

http://hal.inria.fr/inria-00194528

Many reference books, as[START_REF] Ivey | Cartan for beginners: differential geometry via moving frames and exterior differential systems[END_REF], prone the left invariant version rather than the right invariant one used here. In that setting Ω = (τ ) -1 (dτ ).

We can write (6) as D(ῑf )(z) = ῑ(Df )(z) + σ * Kv(g * f ) (z) so that, by [16

) http://hal.inria.fr/inria-00194528 Acknowledgment I am grateful to Gloria Mari-Beffa and Peter Olver. The discussions we have had on the role of the Maurer-Cartan matrix and its significance were determinantal to the present results. This wouldn't have happened hadn't I visited the Institute for Mathematics and its Applications, with additional support from the Fullbright visiting scholar program. I would like to thank IMA, staff and directors who made my year in Minneapolis possible, productive and enjoyable. I have a special thought for Chehrzad Shakiban for both her encouragements and her hospitality.


Catherine Sarret
email: csarret@chu-clermontferrand.fr.

Jean-Jacques Lemaire

Anna Sontheimer

Jérôme Coste

Nadia Savy

Bruno Pereira

Bastien Roche

Odile Boespflug-Tanguy

DTI in PLP1-related

Keywords: PLP1-related disorders, Pelizaeus-Merzbacher disease, Spastic paraplegia type 2, diffusion tensor imaging, tractography, Proteolipid Protein

Aims: We performed quantitative diffusion tensor imaging and brain tractography to distinguish clinical severity in a series of 35 patients with hypomyelinating PLP1-related disorders classified using the Motor Developmental Score according to best motor function acquired before 5 years and the gross motor function measure (GMFM) at the time of MRI acquisition.

Methods:

We calculated fractional anisotropy and diffusivity values in 26 regions of interest, and the numbers of fibers and volumes of hemisphere tractograms. Fiber bundles on tractograms were characterized according to three criteria: size, direction of main-stream fibers, and connectivity of bundles (extratelencephalic projections, commissural fibers, and intrahemispheric connections).

Results: Age-adjusted multivariate analysis in three severity groups revealed increased isotropic diffusion in the superior cerebellar peduncle and grey matter in the most severe group, and larger tractogram volumes and increased numbers of fibers in the least severe group. Tractogram patterns showed preserved extratelencephalic projections and a main anterior-posterior aspect of intrahemispheric fibers in most patients, whereas interhemispheric connectivity was variable. The most severe and intermediate patients had less intrahemispheric connectivity, with a frequent predominant anterior-posterior direction of main-stream fibers.

Interpretation: Diffusion tensor imaging and tractographic parameters can operate as biomarkers to distinguish clinical severity in PLP1-related disorders and could improve our understanding of hypomyelinating leukodystrophies.

Introduction

Pelizaeus-Merzbacher disease (PMD, OMIM312080) and spastic paraplegia type 2 (SPG2, OMIM312920) are X-linked hypomyelinating leukodystrophies caused by mutations in the Proteolipid protein 1 (PLP1) gene that encodes the major central nervous system (CNS) myelin proteins, PLP and DM20 in oligodendrocytes [START_REF] Gow | Conservation of topology, but not conformation, of the proteolipid proteins of the myelin sheath[END_REF][START_REF] Klugmann | Assembly of CNS myelin in the absence of proteolipid protein[END_REF]. The clinical findings associated with PLP1 mutations span a broad-spectrum continuum extending from severe congenital PMD forms to relatively mild late-onset SPG2, leading to the concept of PLP1-related disorders [START_REF] Cailloux | Genotype-phenotype correlation in inherited brain myelination defects due to proteolipid protein gene mutations. Clinical European Network on Brain Dysmyelinating Disease[END_REF][START_REF] Inoue | PLP1-related inherited dysmyelinating disorders: Pelizaeus-Merzbacher disease and spastic paraplegia type 2[END_REF]. We previously classified patients according to the best motor function achieved between the ages of 2 and 5 years to demonstrate a genotype-phenotype correlation [START_REF] Cailloux | Genotype-phenotype correlation in inherited brain myelination defects due to proteolipid protein gene mutations. Clinical European Network on Brain Dysmyelinating Disease[END_REF]. PMD1 (head control) and PMD2 (sitting position) are the most frequent forms and are usually associated with PLP1 duplications.

Missense mutations are rare but usually cause the most severe forms without motor acquisitions and dystonia (PMD0, also named "congenital" forms). Myelin deficiency is the main histological feature in human and mouse models of these severe forms [START_REF] Nave | A single nucleotide difference in the gene for myelin proteolipid protein defines the jimpy mutation in mouse[END_REF][START_REF] Griffiths | Rumpshaker mouse: new X-linked mutation affecting myelination: evidence for a defect in PLP expression[END_REF]. PLP1 loss-of-function (null mutations and large deletions) leads to the mildest PMD3/SPG2 forms with acquisition of walking capacities with/without aid and relatively preserved cognitive development. Murine models and humans lacking PLP1 are characterized by axonal degeneration with a small enlargement of the interlamellar space of the myelin sheath [START_REF] Klugmann | Assembly of CNS myelin in the absence of proteolipid protein[END_REF][START_REF] Garbern | Proteolipid protein is necessary in peripheral as well as central myelin[END_REF]. In all forms, the myelin deficiency is demonstrated by the extensive abnormalities of multimodal evoked potentials and the diffuse hypomyelinated pattern of the supratentorial white matter (WM) magnetic resonance imaging (MRI) [START_REF] Boespflug-Tanguy | Genes involved in leukodystrophies: a glance at glial functions[END_REF]. Most patients clinically improve during childhood, exactly like patients with cerebral palsy. Some progression of myelination is observed until 12 years of age [START_REF] Sarret | Time-course of myelination and atrophy on cerebral imaging in 35 patients with PLP1-related disorders[END_REF]. The second decade is marked by neurological deterioration with cortico-subcortical atrophy on MRI, leading to severe quadriplegia, amyotrophy, optic atrophy and cognitive decline in young adults, even in the mildest forms [START_REF] Schiffmann | An update on the leukodsytrophies[END_REF]. Several recent studies have underlined the correlation between clinical severity and brain atrophy on MRI [START_REF] Sarret | Time-course of myelination and atrophy on cerebral imaging in 35 patients with PLP1-related disorders[END_REF][START_REF] Laukka | Neuroradiologic correlates of clinical disability and progression in the X-linked leukodystrophy Pelizaeus-Merzbacher disease[END_REF][START_REF] Sumida | The magnetic resonance imaging spectrum of Pelizaeus-Merzbacher disease: A multicenter study of 19 patients[END_REF].

Using diffusion tensor imaging (DTI), Laukka et al. found that the WM mean and radial diffusivities of 12 PLP1-mutated patients were higher than controls, in accordance with the severe myelin deficiency observed in PMD [START_REF] Laukka | Diffusion tensor imaging of patients with proteolipid protein 1 gene mutations[END_REF]. Axial diffusivity, which is assumed to be more specific to axon properties, was much less affected, except in the most severe patients. Here we investigated the ability of DTI parameters to distinguish the different severity groups (PMD0-1, PMD2, and PMD3-4) and to correlate with clinical severity in a cohort of 35 PLP1-mutated patients. We extended the WM analysis with tractography in an attempt to identify patterns of WM fiber damage.

Patients and methods

Patients

Thirty-five patients from 32 families with a genetically-confirmed diagnosis of PLP1 mutation (19 duplications, 9 missense mutations and 7 null mutations) were included in this prospective study from January 2004 to December 2009, after first obtaining approval from the institutional review board (CPP Sud-Est VI No. AU788, CNIL No. 1406552, AFSSAPS No. B90298-60) and signed informed consent from the parents. For MRI acquisition, sedation by intrarectal pentobarbital: 3 to 5 mg/kg (maximum dose: 100 mg) and oral alimemazine (4% drops, 1 drop/kg, i.e. 1.25 mg/kg), was performed in patients aged under 6 years old or requiring sedation to avoid movement artifacts.

Our study focused on a comparison of patients with distinct degrees of disease severity. Patients were classified according to the best motor developmental score (MDS) observed between age 2 and 5 years: PMD0, without motor acquisition (n=4 patients); PMD1, head control (n=5); PMD2, sitting position (n=16); PMD3, walking with support (n=8); PMD4, independent walking (n=2). An 88-item gross motor function measure (GMFM) was assessed for all patients at the time of MRI acquisition for motor clinical severity [START_REF] Russell | The gross motor function measure: a means to evaluate the effects of physical therapy[END_REF]. MDS, GMFM, genotype and ages at GMFM and MRI acquisitions are summarized in Tractography was carried on in 24 out of 35 patients at median age 13.6 years (2.3-43.5). For some patients, we could not analyze quantitative DTI (n=3) or tractography (n=11) maps due to corrupted data after MRI acquisition. Data were considered corrupted when DTI and tractography software could not read DICOM series because of artefacts due to subject motion. For eight patients, the quantitative DTI and tractography were analyzed on two different MRI scans at different ages in order to obtain non-corrupted data.

Diffusion imaging and parameters

Images were acquired on a 1.5-Tesla Siemens MR scanner under sedation. DTI acquisition used a single-shot echo-planar dual spin-echo sequence: repetition time (TR) =3600ms, echo time (TE) =85ms, 90° flip angle; six diffusion gradients; b0=600 s/mm²; 35 joined axial slices; 256×256 matrix; voxel size=2×2×6mm³; total acquisition time=6 min. T1-weighted, T2-weighted and fluidattenuated inversion recovery (FLAIR) sequences were added for anatomical guidance (see Supplementary Material 1 for acquisition details).

The DTI parameters fractional anisotropy (FA), mean diffusivity (MD), axial diffusivity (AD) and radial diffusivity (RD) were computed with DTI studio (Jiang and Mori, Johns Hopkins University, cmrm.med.jhmi.edu or godzilla.kennedykrieger.org) using a multivariate linear fitting algorithm for diffusion tensor calculation [START_REF] Basser | MR diffusion tensor spectroscopy and imaging[END_REF]. We explored 26 regions of interest (ROI), 17 in supratentorial and infratentorial WM and 9 in cortical and deep grey matter (GM) (listed in Supplementary Material 2), after automatic correction of eddy current distortion and automatic segmentation of ROI on the diffusion data based on Harvard-Oxford cortical and subcortical structural atlases (FSL, FMRIB Software Library, fsl.fmrib.ox.ac.uk).

Tractograms

Tractography was computed after correction of eddy current distortion, co-registration (mutual information algorithm) and anterior commissure -posterior commissure alignment. Right and left tractograms were generated by a trained neurologist (CS), using fiber assignment by continuous tracking (FACT) [START_REF] Mori | Fiber tracking: principles and strategies -A technical review[END_REF] and tensor deflection (TEND) [START_REF] Lazar | White matter tractography using diffusion tensor deflection[END_REF] methods implemented within iPlan Stereotaxy 3.0 software (BrainLab, Feldkirchen, Germany). FA threshold was set at 0.20 with minimum fiber length tracking of 40 mm. For each tractogram, volume and number of fibers (NOF) were computed and volume/ total intracranial volume (TIV) and NOF/TIV ratios were calculated.

TIV was preferred over brain volume for normalization in order to consider brain growth in children and progressive atrophy in adults. The tractograms were interpreted at individual level and visually compared against two control subjects (aged of 8 and 41 years) by two experienced neurologists (CS and JJL) using the following morphological criteria of fibers bundles: (1) size: large, small, lacking;

(2) direction of main-stream fibers (DMSF): anterior-posterior (longitudinal), lateral-medial (transversal), superior-inferior (vertical); (3) connectivity, irrespective of direction, e.g. fugal or petal: extratelencephalic projections (ETP), commissural fibers (CF) and intrahemispheric connections (IHC).

Statistical analyses

Statistical analysis was performed using Stata 13 software (Stata-Corp, College Station, TX). The tests were two-sided, with a type-I error set at α=0.05. Quantitative data, expressed as mean and standard deviation or median and min-max ages according to statistical distribution, were compared between three independent groups (PMD0-1, PMD2, and PMD3-4) using a Kruskal-Wallis test followed by the appropriate post-hoc Dunn's test. ANCOVA was performed with age as the adjustment covariate. The assumption of the normality of residuals was also studied using the Shapiro-Wilk test. Data exhibiting non-normal distributions, a log-transformation was proposed to achieve normality of several variables and to allow the correct use of these analyses. We chose to report all the individual p-values without doing any mathematical correction for distinct tests comparing groups [START_REF] Rothman | No adjustments are needed for multiple comparisons[END_REF]. A focus was given to the magnitude of improvement and to the clinical relevance [START_REF] Feise | Do multiple outcome measures require p-value adjustment?[END_REF]. Results were expressed as effect-sizes and 95% confidence intervals, and can be compared to the effect-size bounds proposed by Cohen [START_REF] Cohen | Statistical power analysis for the behavioral sciences[END_REF], i.e. small (±0.2), medium (±0.5) and large (±0.8, "grossly perceptible and therefore large"). Finally, a nonparametric Spearman's correlation test, with Sidak's type-I error correction, was used to compare individual quantitative DTI parameters and GMFM.

Results

Patient severity

The best motor developmental score (MDS) of the 32 patients analyzed with quantitative DTI were classified as follows: PMD0-1 (n=8), PMD2 (n=14) and PMD3-4 (n=10), with median ages of The level of the motor handicap at the time of the MRI acquisition was evaluated in the 32 patients using the gross motor function measure (GMFM). Comparison of mean GMFM adjusted to the age between the three MDS groups showed that GMFM was higher in PMD3-4 than PMD0-1 and PMD2. It was also higher in PMD2 than PMD0-1 but not significantly (Fig. 1).

Quantitative diffusion parameters

Results of FA, MD, AD, RD in 17 ROI of WM and 9 ROI of GM are shown in Figure 2 (see Supplementary Table 1 for values and Supplementary Fig. 1 for effect-sizes and 95% confidence intervals). We found that FA was significantly lower in PMD2 than PMD3-4 in middle cerebellar peduncle, in PMD0-1 than PMD2 in middle temporal gyrus, right caudate nucleus and right thalamus, and in PMD0-1 than PMD3-4 in postcentral gyrus. MD was significantly higher in PMD0-1 than PMD2 in superior cerebellar peduncle, frontal pole and left caudate nucleus, and in PMD0-1 than PMD3-4 in superior cerebellar peduncle. AD was significantly higher in PMD0-1 than PMD2 in superior cerebellar peduncle, left posterior and right superior corona radiata, frontal pole and left caudate nucleus, and in PMD0-1 than PMD3-4 in left superior cerebellar peduncle and right superior corona radiata. RD was significantly higher in PMD0-1 than PMD2 in superior cerebellar peduncle, frontal pole, middle temporal gyrus, caudate nuclei and thalami, and in PMD0-1 than PMD3-4 in right superior cerebellar peduncle and right caudate nucleus.

Motor abilities, evaluated by the GMFM performed at the time of MRI, showed a relatively strong negative linear correlation with MD (p<0.043) and AD (p<0.036) in the superior cerebellar peduncles. Other possible strong linear relations were observed with MD, AD or RD in anterior corona radiata but were not statistically significant (Fig. 3; Supplementary Table 2).

Tractograms

Volume/TIV and NOF/TIV ratios were higher in PMD3-4 than in other groups (p<0.02) after adjustment to age, considering an older age of PMD3-4 patients at MRI acquisition (Fig. 4, Supplementary Table 3). ETP appeared relatively preserved in all groups. CF were variably individualized in all groups, ranging from large fibers (n=9) to small fibers (n=8) or a lack of connectivity (n=7). IHC size was clinical severity-dependent, being clearly smaller (n=8) or lacking connectivity (n=4) in PMD2 and PMD0-1 patients. DMSF of IHC was analyzable in 9 out of 9 PMD3-4 patients and 10 out of 12 PMD2 patients, and analysis showed major tracking of longitudinal fibers in 18 out of 19 patients whereas one patient had major transversal fibers (Fig. 5A; Supplementary Fig. 2 and3). There were age-dependent differences in the PMD2 group, with smaller CF and IHC in the youngest and oldest patients while patients in late childhood had the largest projections (Fig. 5B; Supplementary Fig. 2 and3).

Discussion

Here we analyzed quantitative DTI measures and tractography patterns from acquisitions performed at the same medical center in a large cohort of PLP1-mutated male patients in order to investigate the ability of these technics to distinguish severity groups. Our 35 patients were representative of the broad clinical spectrum observed in PLP1-related disorders: 46% classical PMD2 forms, 26% severe PMD0-1 forms, and 28% mild PMD3-SPG2 forms. For clinical classification, we used the MDS previously demonstrated to correlate to genotype [START_REF] Cailloux | Genotype-phenotype correlation in inherited brain myelination defects due to proteolipid protein gene mutations. Clinical European Network on Brain Dysmyelinating Disease[END_REF]. To correlate motor skills at the time of MRI, we used GMFM [START_REF] Russell | The gross motor function measure: a means to evaluate the effects of physical therapy[END_REF], a validated clinical score in cerebral palsy that is widely used in the quantification of motor abilities in leukodystrophic patients. Comparison of mean GMFM between MDS groups argued as in CP for a correlation between the level of the best motor development always observed before 5 years of age and the severity of the motor disabilities later in life.

This study shows that quantitative DTI parameters in supratentorial WM are insufficient to distinguish clinical severity in PLP1-mutated patients, whereas increased isotropic diffusion in cortical and deep GM and in infratentorial WM was clinical severity-linked. We also found clinical severity-dependent differences on quantitative and descriptive tractography.

Laukka et al. previously identified increased MD and RD and decreased FA in supratentorial WM regions in 12 PMD patients compared to controls as expected in hypomyelinating brains [START_REF] Laukka | Diffusion tensor imaging of patients with proteolipid protein 1 gene mutations[END_REF].

Steenweg et al. also found that RD in supratentorial WM was inversely correlated with motor handicap in a cohort of 28 patients with various hypomyelinating disorders [START_REF] Steenweg | Magnetic resonance imaging pattern recognition in hypomyelinating disorders[END_REF]. Here, comparison of the MDS groups of our PLP1-mutated patients failed to differentiate clinical severity using quantitative DTI parameters in supratentorial WM. A previous study in the same cohort of patients was unable to classify clinical severity using a myelination score on conventional MRI [START_REF] Sarret | Time-course of myelination and atrophy on cerebral imaging in 35 patients with PLP1-related disorders[END_REF]. Moreover, motor abilities at the time of MRI using GMFM did not significantly decrease with increasing diffusion in supratentorial WM. These results confirmed that increased diffusion and decreased myelination score on the supratentorial WM demonstrate the myelin defect but may not serve to classify the clinical severity observed in PLP1-related disorders. Factors that may have limited the ability of this study to identify differences include the small sample sizes in each group, the broad age-ranges in the patients, or the algorithms and tools used in the quantitative analyses.

Other factors could be an offset between microarchitectural changes and clinical features.

Progression in both motor acquisitions and myelination are observed in the first decade of life but followed by motor and cognitive decline and progressive cerebral atrophy [START_REF] Sarret | Time-course of myelination and atrophy on cerebral imaging in 35 patients with PLP1-related disorders[END_REF], so microarchitectural changes in supratentorial WM may precede clinical changes, creating a time gap between diffusion data and clinical data.

In infratentorial WM, clinical severity of the MDS groups and motor abilities at the time of MRI are in accordance with the DTI parameter values, except FA. This result was surprising, as infratentorial WM is usually well myelinated on structural MRI in PLP1-related disorders regardless of clinical severity [START_REF] Feise | Do multiple outcome measures require p-value adjustment?[END_REF]. However, Laukka et al. also found changes in MD and RD in cerebellar peduncle of PMD patients compared to controls [START_REF] Laukka | Diffusion tensor imaging of patients with proteolipid protein 1 gene mutations[END_REF]. Moreover, infratentorial abnormalities had been described in a DTI study by Hassen et al. that found MD, AD and RD variations in spinal cord of a SPG2 mouse model [START_REF] Hassen | Characterisation of spinal cord in a mouse model of spastic paraplegia related to abnormal axono-myelin interactios by in vivo quantitative MRI[END_REF]. These results suggest that PLP1-mutated patients present subtle changes in infratentorial WM microarchitecture. This idea is consistent with (i) the early and predominant signs of cerebellar dysfunctions observed in PMD patients, (ii) the correlation we had previously found between clinical severity and cerebellar atrophy [START_REF] Sarret | Time-course of myelination and atrophy on cerebral imaging in 35 patients with PLP1-related disorders[END_REF], and (iii) neuropathological analysis of PMD patients with cerebellar abnormalities [START_REF] Sima | Neuronal loss in Pelizaeus-Merzbacher disease differs in various mutations of the proteolipid protein 1[END_REF]. In this study, we found higher AD and RD in the most severe patients. Laukka et al. also found changes in AD, which is supposed to reflect axonal integrity, in only the most severe patients, whereas RD, which may reflect myelin integrity, was increased in all forms [START_REF] Laukka | Diffusion tensor imaging of patients with proteolipid protein 1 gene mutations[END_REF].

In cortical and deep GM, clinical severity of MDS groups increased with the increase of diffusion (MD, AD and RD) in several regions. Moreover, diffusion was more isotropic (decreased FA) in GM in most severe patients. To date in PLP1-related disorders, there are no reports of DTI analysis of GM but Laukka et al. and our team already showed a clear cerebral atrophy including GM compartment and correlating with clinical severity [START_REF] Sarret | Time-course of myelination and atrophy on cerebral imaging in 35 patients with PLP1-related disorders[END_REF][START_REF] Laukka | Diffusion tensor imaging of patients with proteolipid protein 1 gene mutations[END_REF]. Hassen et al. also found MD increase in GM of spinal cord in a SPG2 mouse model [START_REF] Hassen | Characterisation of spinal cord in a mouse model of spastic paraplegia related to abnormal axono-myelin interactios by in vivo quantitative MRI[END_REF]. Increased diffusion in cerebral GM has been reported in multiple sclerosis, where it appears to correlate to motor and cognitive decline [START_REF] Cavallari | Microstructural changes in the striatum and their impact on motor and neuropsychological performance in patients with multiple sclerosis[END_REF][START_REF] Calabrese | Cortical diffusion-tensor imaging abnormalities in multiple sclerosis: a 3-year longitudinal study[END_REF],

and in an animal model of genetic demyelinating disorder [START_REF] Bradbury | Diffusion tensor imaging analysis of the brain in the canine model of Krabbe disease[END_REF]. However, contrary to our patients, isotropic diffusion decreased (increased FA) in the cortex in multiple sclerosis as a probably result of local activation and proliferation of microglia characterized by the loss of ramifications and acquiring a more anisotropic, bipolar oriented structure as observed in active acute and chronic cortical lesions [START_REF] Calabrese | Cortical diffusion-tensor imaging abnormalities in multiple sclerosis: a 3-year longitudinal study[END_REF]. In PLP1-related disorders, it may corroborate with (i) the neuron loss, demyelination and axonal damage reported on neuropathological analysis of PLP1-mutated animal models [START_REF] Miller | Proteolipid protein gene mutation induces altered ventilatory response to hypoxia in the myelin-deficient rat[END_REF] and PMD/SPG2 patients [START_REF] Sima | Neuronal loss in Pelizaeus-Merzbacher disease differs in various mutations of the proteolipid protein 1[END_REF], (ii) the absence of local microglial activation and (iii) the neuronal PLP1 expression found in mouse and humans [START_REF] Bongarzone | Identification of a new exon in the myelin proteolipid protein gene encoding novel protein isoforms that are restricted to the somata of oligodendrocytes and neurons[END_REF][START_REF] Sarret | Novel neuronal proteolipid protein isoforms encoded by the human myelin proteolipid protein 1 gene[END_REF]. These results underline that GM is clearly involved in the severity of PLP1-related disorders.

Tractography was possible in PLP1-related disorders, including severe forms. Quantitative parameters including tractogram volumes and number of fibers, which have already been used in previous studies [START_REF] Vassal | Combined DTI tractography and functional MRI study of the language connectome in healthy volunteers: extensive mapping of white matter fascicles and cortical activations[END_REF][START_REF] Wang | The involvement of supratentorial white matter in multiple system atrophy: a diffusion tensor imaging tractography study[END_REF], permitted to distinguish the mildest forms from other forms. Descriptive analyses of fiber bundles showed a preservation of ETP and an altered organization of intra-and interhemispheric connections in all patients, mostly with a loss of vertical and transversal connectivity and an apparent over-representation of longitudinal fibers. This suggests that tractography assessment is easy in mono-directional and partially myelinated fibers but remains difficult in multi-directional and less myelinated fibers in hypomyelinating disorders. Descriptive analyses also revealed an age-dependent evolution of the tractograms in the largest classicalintermediate group. For the youngest patients, the increasing size of tractograms may correspond to different stages of myelination. While healthy 2-year-old children have already achieved myelination except in subcortical fibers, severe supratentorial hypomyelination is the main feature in PLP1related disorders [START_REF] Schiffmann | An update on the leukodsytrophies[END_REF] where myelination can partially improve until 12 years of age [START_REF] Sarret | Time-course of myelination and atrophy on cerebral imaging in 35 patients with PLP1-related disorders[END_REF]. For adult patients presenting severe cerebral atrophy on conventional MRI, the decreased size of tractograms may reflect WM atrophy or loss of connectivity. Laukka et al., argued prominent WM atrophy in PMD/SPG2 patients [START_REF] Laukka | Neuroradiologic correlates of clinical disability and progression in the X-linked leukodystrophy Pelizaeus-Merzbacher disease[END_REF], whereas we observed parallel WM/GM atrophy over time on conventional MRI [START_REF] Sarret | Time-course of myelination and atrophy on cerebral imaging in 35 patients with PLP1-related disorders[END_REF]. Taken together, the combination of patients' clinical evolution, progressive global atrophy, GM microstructure changes on DTI and progressive WM fiber disappearance on tractography converge to confirm that PLP1-related disorders are global CNS diseases affecting both WM and GM.

This study has several limitations. First, recent studies suggested the potential for seemingly minor alterations in data acquisition, in particular the b-value and number of diffusion gradient directions, to impact the DTI parameters [START_REF] Bisdas | Reproducibility, interrater agreement, and age-related changes of fractional anisotropy measures at 3T in healthy subjects: effect of the applied b-value[END_REF]. Moreover, using multiple b-values such as recent technique called neurite orientation dispersion and density imaging (NODDI) could provide additional information on tissue compartments [START_REF] Zhang | NODDI: practical in vivo neurite orientation dispersion and density imaging of the human brain[END_REF]. Further studies should consider these parameters. Second, with only 6 gradient directions, tractography can be limited to only major fiber tracts and can underestimate the connectivity. Our goal to differentiate severity in PLP1-related disorders was achieved in this protocol but it would be interesting to increase the number of directions to study weak connections.

Finally, even if we took into account the age of the subjects as a covariate, there is a difference of age between the patients' groups that could generate a bias of interpretation. We did not report intraindividual evolution of DTI parameters and tractography. This would be a different study that could help to strengthen our results.

In conclusion, this study shows for the first time that tractography can be a useful and non-invasive technique in investigations and follow-up of hypomyelinating disorders. Our findings also bring further evidence of physiopathological mechanisms in PLP1-related disorders that likely involve both grey and white matter and infratentorial structures. These findings must be confirmed in further studies and may be important in helping to develop new therapeutic strategies targeting not only oligodendrocytes but also axons and neurons.

  8.6 years (3.5-17.4), 6.0 years (2.3-19.1) and 15.4 years (5.5-41.0), respectively. Tractography analysis were possible in 3 PMD0-1 (median ages of 8.5 years (3.0-17.4)), 12 PMD2 (median ages 7.7 years (2.3-43.5)) and 9 PMD3-4 (median age 17.4 years (10.8-37.0)) patients.

Figure 1 .

 1 Figure 1. Comparison of mean GMFM between the three MDS groups. Y-axis: GMFM (Gross Motor Function Measure) includes 88 items (best motor score = 88) at the time of MRI acquisition. X-axis: MDS (Motor developmental score) corresponds to the best motor function achieved between the ages of 2 and 5 years. Thirty-two patients are divided in three groups of MDS: PMD0-1 (n=8), PMD2 (n=14), PMD3-4 (n=10). Results are given using ANCOVA with age as the adjustment covariate. * indicates a significant difference (p-value <0.05). p = p-value. Standard deviations are given as an error bar.

Figure 2 .

 2 Figure 2. Comparison of quantitative DTI parameters between the three MDS groups in white matter, subcortical white matter and grey matter. Successive results for fractional anisotropy (FA), mean diffusivity (MD), axial diffusivity (AD) and radial diffusivity (RD). Thirty-two patients are divided in three groups of MDS: PMD0-1 (n=8), PMD2 (n=14), PMD3-4 (n=10). MD, AD and RD values are expressed as 10 -3 values. Results are expressed as means. Standard deviations are given as an error bar. * for p-value <0.05. p-values were computed using ANCOVA with age as the adjustment covariate. ALIC: anterior limb of the internal capsule, PLIC: posterior limb of the internal capsule, R: right, L: left, CC: corpus callosum, Mid: middle, Ant: anterior, Post: posterior, Sup: superior, Ped: peduncle, Cereb: cerebellar.

Figure 3 .

 3 Figure 3. Coefficients of correlation between GMFM and quantitative DTI parameters in white matter, subcortical white matter and grey matter. Using a nonparametric Spearman's correlation test, with Sidak's type-I error correction, values of coefficients of correlation between -1 and 1 are represented graphically with a color-coded heatmap in each region of interest and for each DTI parameters. GMFM: gross motor function measure, FA: fractional anisotropy, MD: mean diffusivity, AD: axial diffusivity, RD: radial diffusivity, ALIC: anterior limb of internal capsule, PLIC: posterior limb of internal capsule, R: right, L: left, CC: corpus callosum, Mid: middle, Ant: anterior, Post: posterior, Sup: superior, Ped: peduncle, Cereb: cerebellar.

Figure 4 .

 4 Figure 4. Comparison of quantitative tractography parameters between the three MDS groups. Successive results for total intracranial volume (TIV), right and left tractogram volume / TIV and right and left tractogram number of fibers / TIV. Twenty-four patients whom tractography was available are divided in three groups of MDS: PMD0-1 (n=3), PMD2 (n=12), PMD3-4 (n=9). Results are expressed as means. Standard deviations are given as an error bar. Total intracranial

Figure 5 .

 5 Figure 5. Examples of frontal views of both hemisphere tractograms and morphological analyses of tractographies of different MDS groups and ages in PMD/SPG2 patients. Tractograms were realized for two regions of interest corresponding to the right/left hemispheres. Colors correspond to the direction of main-stream fibers (DMSF): blue = vertical direction, green = longitudinal direction, red = transversal direction. For clinical comparison, Gross Motor Functional Measure (GMFM) at the time of MRI is given under each tractogram when available. A. Examples of tractograms for each group of severity and table of the results of morphological analyses in 24 patients whom tractography was available: PMD0-1 (n=3), PMD2 (n=12) and PMD3-4 (n=9).Extratelencephalic projections (ETP) were relatively preserved in all groups, but were smaller than in controls. Compared to controls (a,e), commissural fibers (CF) and intrahemispheric connections (IHC) were smaller or lacking in all groups: (1) the most severe PMD0-1 (b,f), (2) the classical PMD2 (c,g) and the mildest PMD3-4 patients (d,h). DMSF in IHC was marked by longitudinal (or rare transversal) over representation. B. Each image corresponds to tractograms of PMD2 patients of different ages. These results show that (1) the youngest patients (a,b) had less-developed ETP, CF and IHC than patients at the late infancy (c,d) and adolescent patients (e,f); (2) the oldest patients (g,h) had smaller ETP, CF and IHC than adolescent patients (e,f).

Table

  . At the first MRI, median age was 8.6 years (2.3 to 43.5). Quantitative DTI measures were obtained in 32 out of 35 patients at median age 8.6 years (2.3 to 41.0).

Table 1 . List of patients and families with PMD/SPG2

 1

	Patient	Family	MDS	Age at DTI acquisition +/-GMFM	GFMF88 at MRI acquisition	PLP1 mutation	Family genetic number*
				(years)
	1	1	2	2.3	29.3	duplication	1910
	2	2	0	3.0	NA	c.454-1G>A	767
	3	3	1	3.5	36.8	c.327delC	1171
	4	4	2	3.8	25.3	duplication	NA
	5	5	2	4.3	34.8	duplication	1251
	6	6	2	4.4 / 6.8	26 / 36.8	duplication	1059
	7	7	2	4.5 / 7.5	NA / 39.5	c.659G>A	759
	8	8	2	4.7	NA	duplication	1201
	9	9	2	5.1	38.4	duplication	1026
	10	10	3	5.5	39	duplication	1114
	11	11	1	5.7	28.7	duplication	968
	12	12	2	5.7	26.7	duplication	887
	13	13	2	6.2	47.1	c.125delG	1038
	14	14	3	8.3 / 11.5	44.2 / NA	duplication	407
	15	15	0	8.5	20.5	duplication	545
	16	16	1	8.6	17	duplication	1776
	17	17	0	8.6	NA	c.670C>T	766
	18	18	2	8.6	41.4	duplication	418
	19	19	3	10.8 / 13.3	NA / 45	c.173A>G	106
	20	20	1	12.3	NA	duplication	440
	21	21	3	13.5	NA	duplication	46
	22	22	2	13.6	46.9	duplication	154
	23	23	3	14.5 / 17.4	49.9 / 30.5	c.307delG	905
	24	24	2	14.8	40	duplication	61
	25	25	1	15.0	NA	duplication	150
	26	26	4	16.2	56.6	c.548C>A	41
	27	27	2	16.8 / 19.1	0 / 0	c.608A>G	37
	28	28	2	17.0	24	duplication	28
	29	23	3	17.2 / 20.1	44.4 / 32.3	c.307delG	905
	30	29	0	17.4	NA	c.454-1G>T	26
	31	30	3	19.0	24	c.646C>G	53
	32	31	2	21.8	NA	c.646C>T	24
	33	19	3	32.8	NA	c.173A>G	106
	34	32	4	37.0 / 41.0	31.8	c.801G>T	1
	35	7	2	43.5	NA	c.659G>A	759
	MDS: motor developmental score, age (years) at DTI acquisition and GMFM scoring, GMFM: gross
	motor function measure realized at the time of MRI acquisition, and genotype (at the DNA level).
	*Family genetic number corresponds to our laboratory family number and refers to previous
	publication (Cailloux et al. 2000). NA : not available.

DTI in PLP1-related disorders Sarret et al.

Acknowledgements

The authors thank the members of the Leukofrance network who send us the patients (MA Barthez-Carpentier, F Boidin, P Burkart, S Ceylaner, B Chabrol, D Chaigne, G Cioni, B Echenne, M Elleder, P Evrard, M Garcia Silva, K Kluger, J M Lopez-Terradas, J Motte, A Munnich, D Nicholls, N Philip, G Ponsot, T Reckert, D Rodriguez, C Rousselle, G Sébire, H Steinbock, M Troncoso, H Van Esche, and L Van Maldergem) and Dr JP Renou for precious help with the DTI analyses.

Funding

This study received funding from the ELA foundation, the FP7-RD Connect, the national PHRC [hospital-based clinical research program] for Clermont-Ferrand university hospital, and the Région Auvergne regional council. JJL, CS, AS, JC, NS report a team non-financial support from BrainLab. The other authors declare no competing interests.

List of Supplementary Material