Samir Bekkara
email: samir.bekkara@gmail.com

Abdelghani Zeghib
email: zeghib@umpa.ens-lyon.fr

SINGULAR RIEMANNIAN METRICS, SUB-RIGIDITY VS RIGIDITY

Keywords:

published or not. The documents may come

1.0.1. A correspondence: contact sub-Riemannian → Riemannian. We will henceforth assume that D is a contact hyperplane, i.e. ω 0 ∧ (dω 0 ) d is a volume form where dim M = n = 2d + 1. Let us recall the (classical) construction of a natural Riemannian metric h associated to (D, h). Observe that dω 0 is a symplectic form on D and that for a function f , d(f ω 0 ) = f dω 0 (on D). Assume D orientable, and let α be the Riemannian volume form derived from h on it. Writing that dω d = α on D determines uniquely f , in other words the Riemannian metric (together with an orientation) allows one to choose a canonical contact form, say ω 0 . Let R be the Reeb field of ω 0 : i R dω 0 = 0, and ω 0 (R) = 1. We extend the metric h to a metric h on T M by declaring that R is unit and orthogonal to D. The orientation is actually irrelevant since the inverse orientation gives the same metric.

Isometry groups of Lie type. Summarizing up, a contact sub-Riemannian metric generates a Riemannian metric. In particular the isometry group Iso(M, h) is a (closed) sub-group of Iso(M, h). Similarly for pseudo-groups of local isometries (i.e. isometries defined between open sets of M , and composed when this is possible). It then follows that the isometry group of (M, h) as well as its local isometry pseudo-group are of Lie type (of finite dimension).

CARTAN'S FINITE TYPE CONDITION. (see [START_REF] Kobayashi | Transformation groups in differential geometry[END_REF]) Let H be a subgroup of GL n (R). An H structure on a manifold M n is a reduction of the structural group of its frame bundle GL (1) (M ) = P (M ) to H. Equivalently, this is a section of P (M )/H (assume here to simplify that H is closed). A Riemannian metric corresponds to a O(n)-structure. A sub-Riemannian metric corresponds to an H structure where H is the subgroup of GL n (R) preserving R n-1 and the standard Euclidean product on it. Its elements have the form:

A -→ u 0 b
where A ∈ O(n -1), -→ u ∈ R n-1 and b ∈ R.

Following Cartan, one associates to H its k-prolongation H k , a space of symmetric (k + 1)-multi-linear forms on R n with values in R n . If A ∈ H k , and v 1 , . . . , v k ∈ R n are fixed, then v → A(v, v 1 , . . . , v k ) belongs to End(R n ). By definition of a prolongation, the last element is assumed to be in the Lie algebra of H.

An H-structure has a finite type k ∈ N, if H k = 0. The principal result of Cartan theory is that the isometry group of an H-structure of finite type is a Lie group. The remarkable fact here is that being of finite type depends only on H (as a subgroup of GL n (R)) and not on the structure itself. As an example, a sub-Riemannian metric has infinite type, no matter it is integrable or contact! If fact, the test of finiteness of type concerns the case of the flat translation-invariant Hstructure on R n . The flow of a vector field V preserves this structure iff, for any x ∈ R n , the derivative D x V belongs to the Lie algebra H. The (k + 1)-coefficient in the Taylor development in a linear coordinates of V belongs to H k . Hence, finite type means V is polynomial.

GROMOV'S RIGIDITY. (see [START_REF] Ballmann | Geometric structures[END_REF][START_REF] Benoist | Orbites des structures rigides (d'après M. Gromov)[END_REF][START_REF] Candel | Parallelisms, prolongations of Lie algebras and rigid geometric structures[END_REF][START_REF] Gromov | Lectures on transformation groups: geometry and dynamics[END_REF][START_REF] Gromov | Rigid transformations groups[END_REF]) Gromov's definition of geometric structures consists essentially in giving up the "infinitesimal homogeneity" in the case of Cartan's H-structures. As examples, functions, vectors fields are geometric structures, and also is a " finite union" of geometric structures. Isometries are defined naturally. Gromov introduces a rigidity condition which coincides with finiteness of type in the case of H-structures.

Rough-definition.

If σ is such a structure on a manifold M , and x ∈ M , let Iso Loc

x (σ) be the group of (germs) of isometries defined in a neighborhood of x and fixing x. For an integer k, denote by Diff k x (M ) the group of k-jets of diffeomorphisms of M fixing x. We have a map jet k

x : Iso Loc x (σ) → Diff k x (M ). The intuitive idea of rigidity (of order k) is that jet k

x is injective: an isometry is fully determined by its k-jet. We say in this case that σ is Iso-rigid at order k. For example, a Riemannian metric is Iso-rigid at order 1: an isometry is determined by giving its derivative at some point. In the case of sub-Riemannian metrics, jet 1

x is injective in the contact case (since it generates a Riemannian metric), but for no k, jet k

x is injective in the integrable one.

Definition. We then conclude a divergence between this intuitive formalization of rigidity and Cartan's finiteness of type. The true Gromov's definition is actually of infinitesimal nature. For an H-structure σ, one defines the group Iso k+1 x (σ) ⊂ Diff k+1

x (M ) as the group of (k + 1)-jets of diffeomorphisms preserving σ up to order k at x. For example, if σ is a Riemannian metric, then a (local)-diffeomorphism f gives rise to a (k + 1)-infinitesimal isometry ∈ Iso k+1

x (σ) if f * σσ vanishes up to order k at x. (In the general case of a geometric structure σ of order r, f is a (k + r)-isometry if f * σ and σ have the same k-jet at x). The true definition of k-rigidity is that jet k : Iso k+1

x (σ) → Iso k x (σ) is injective for any x.

Example. Let us see how this injectivity default happens in the example of the contact form ω = dz + xdyydx on R 3 , endowed with the restriction of dx 2 + dy 2 . It corresponds to a left invariant contact sub-Riemannian structure on the Heisenberg group, and hence it is homogeneous. Consider f : (x, y, z) → (x + δ(z), y + δ(z), z). Assume δ(0) = 0, then, f (0) = 0. Thus, jet k+1 0 f determines a (k + 1)-isometry at 0, iff, ∂δ ∂z k (0) = 0 or equivalently jet k 0 (f ) = 1 (that is f has the same jet as the identity). So any such δ with a non-trivial ∂δ ∂z k+1 (0) = 0 determines an isometry violating the injectivity of Iso k+1 → Iso k .

Remark. Gromov's definition strictly coincides with finiteness of type in the case of H-structure (see [START_REF] Ballmann | Geometric structures[END_REF], Example 3.17), and thus sub-Riemannian metrics are krigid for no k.

Let us end this criticism on the rough definition Iso-rigidity by noting that a generic geometric structure (e.g. a Riemannian metric) has no non-trivial local isometries, in which case the local rigidity condition is empty. In contrast, it is the infinitesimal rigidity condition (even empty) that allows one to associate "rigid" (solid!) objects to σ, independently of the fact that it has or not local isometries. Indeed, it is proved in both Cartan and Gromov situations, that k-rigidity (or kfiniteness of type) allows one to construct a parallelism canonically associated to σ defined on the k-frame bundle GL (k) (M ) → M (this is the usual frame bundle for k = 1) (see for instance [START_REF] Ballmann | Geometric structures[END_REF] and Theorem 2 in [START_REF] Candel | Parallelisms, prolongations of Lie algebras and rigid geometric structures[END_REF]). This produces in particular the Levi-Civita connection and hence geodesics for Riemannian metrics. Also, the Lie group property is proved by means of this framing.

LIGHTLIKE METRICS

Duality. Our original motivation was to study rigidity of lightlike metrics. They are simply H-structures where H consists of matrices:

b -→ u 0 A where A ∈ O(n -1), -→ u ∈ R n-1 and b ∈ R.
Observe that this is exactly the dual of the group defining sub-Riemannian metrics, that is, the automorphism A ∈ GL n (R) → A * -1 ∈ GL n (R) sends one group onto the other.

More geometrically, one defines a lightlike scalar product on a vector space as a positive symmetric bilinear form having a kernel of dimension one. A lightlike metric on a manifold M is a tensor which is a lighlike scalar product on each tangent space. More generaly, a lighlike metric on a vector bundle E → M consists in giving a 1-dimensionnal sub-bundle N ⊂ T M together with a Riemannian metric on E/N . If one defines a sub-Riemannian metric on E → M as a codimension 1 sub-bundle D ⊂ E endowed with a Riemannian metric, then one gets a duality: lightlike metric on E ↔ sub-Riemannian metric on (the dual) E * . In other words, a lightlike metric g on a manifold M consists in giving a line subbundle (direction field) N ⊂ T M , and a Riemannian metric on T M/N . The direction field N and the 1-dimensional foliation N that it generates are called characteristic.

Natural situations. Lightlike metrics appear naturally as induced metrics on submanifolds of Lorentz manifolds. Indeed let (L, h) be a Lorentz manifold, and M ⊂ L a submanifold such that for any x ∈ M , the restriction h x on T x M is degenerate. Then, this is a lightlike metric on M , i.e. h x has a kernel of dimension 1 and is positive on T x M . As an example, by definition characteristic hypersurfaces of the D'Alembertian operator on L are lightlike hypersurfaces. Also, horizons (in particular of black holes if any) of subsets of L are topological hypersurfaces and are lightlike whence they are smooth. Now we give two opposite classes of examples of lightlike metrics which correspond, by duality to the integrable and contact cases of the sub-Riemannian situation, respectively:

Transversally Riemmannian lightlike metrics. A lightlike metric on a manifold I of dimension 1 is just 0. Consider now a direct product of (I, 0) with a Riemannian metric (Q, h). This gives a lightlike metric h ⊕ 0 on Q × I. A lightlike metric g on a manifold M is called transversally Riemannian if it is locally isometric to such a (direct) product. This is equivalent to the fact that the flow of any vector field tangent to the characteristic direction N preserves g (it suffices that this happens for one non-singular such vector field).

Generic lightlike metrics. Let X be a vector field tangent to N . One sees that N annualizes the Lie derivative L X g (i.e. L X g(u, v) = 0, if u ∈ N ). Furthermore, L X g is conformally well defined: if X ′ is another vector field tangent to N , then L X ′ g = f L X g, for some function f on M . We say that g is generic if L X g has maximal rank, i.e. its kernel is exactly N . This therefore defines a conformal pseudo-Riemannian structure on T M/N .

Rigidity flavours.

Exactly as in the sub-Riemannian case, lightlike metrics have infinite type and thus are not rigid. Indeed, the local isometry group has infinite dimension for any transversally-Riemannian metric. For example, if

M = R n-1 × R, with the metric dx 2 1 +. . . dx 2 n-1 , then any map f (x, t) = (x, l(x, t)) is isometric.
The key observation of [START_REF] Bekkara | Actions of semisimple Lie groups preserving a degenerate Riemannian metric[END_REF] was a kind of Liouville theorem for the lightcone Co n , n ≥ 3. This is R + × S n-1 endowed with the lightlike metric g (t,x) = e 2t Can x , where Can is the usual metric on S n-1 . This is in fact the lightcone at 0 in the Minkowski space M in n+1 . The lorentz group O + (1, n) acts isometrically on M in n+1 and hence on Co n . The observation is that any local isometry of Co n coincides with the restriction of an element of O + (1, n).

A correspondence: "generic transversally conformal Lightlike geometry" ↔ Conformal Riemannian geometry. The cone situation generalizes to that of transversally conformal lightlike structure. This means that the flow of any X tangent to N is conformal, equivalently L X g = f g for some function f . Locally, M = Q × I where I is an interval, and g (q,r) = c(q, r)h q , where h is a Riemannian metric on Q.

Assume φ is an isometry of (M, g), then it acts on Q, the quotient space of its characteristic foliation and induces a diffeomorphism ψ, which is obviously conformal for (Q, h).

Conversely, let ψ a conformal transformation of (N, h), and let us look for an isometry of (M, g) of the form φ : (q, r) → (φ(q), δ(q, r)). We assume here g generic, which means that ∂c(q,r) ∂r = 0. Let us assume that I = R, and for any q, the map r → c(q, r) is a global diffeomorphism of R. If f is the conformal distortion of ψ, that is ψ * h = f h, then φ is isometric iff c(φ(q), δ(q, r))f (q) = c(q, r). Our hypotheses imply that for any fixed q, δ(q, r) can be uniquely chosen, and hence a conformal transformation of (N, h) admits a unique isometric lifting on (M, g). (One may compare with a somehow similar construction in [START_REF] Fefferman | Conformal invariants" The mathematical heritage of Élie Cartan[END_REF]).

SUB-RIGIDITY OF GEOMETRIC STRUCTURES, RESULTS

We have the following infinitesimal result for lightlike metrics (where jet k x (φ) = 1, means that φ has the same k-jet as the identity at x): Theorem 3.1. Let g be a generic lightlike metric on a manifold M of dimension n ≥ 4. Then, a 3-infinitesimal isometry with a trivial 1-jet, has a trivial 2-jet:

φ ∈ Iso 3 x , jet 1 x (φ) = 1 =⇒ jet 2 x (φ) = 1
This notion was actually brought out by Benveniste-Fisher in [START_REF] Benveniste | Nonexistence of invariant rigid structures and invariant almost rigid structures[END_REF] under the name of almost-rigidity. We believe here that the word "sub-rigid" is more telling (see also [START_REF] Dumitrescu | Meromorphic almost rigid geometric structures[END_REF]).

In order to keep an elementary level of exposition, we restrict ourselves to geometric structures of order 1, that is, GL n (R)-equivariant maps P (M )(= GL (1) (M )) → Z, where P (M ) is the frame bundle of M and Z is a manifold with a GL n (R)action. The classical case of an H structure corresponds to the homogeneous space

Z = GL n (R)/H. Definition 3.2. A geometric structure σ is (k + s, k)-sub-rigid, if any (k + s)- isometry whose k-jet is trivial has a trivial (k + 1)-jet; formally, if Im k+s,k+1
x denotes the image of Iso k+s x → Iso k+1

x , then, for any x, Im k+s,k+1

x

→ Iso k x is injective. Remarks 3.3. 1. (k + 1, k)-sub-rigidity means k-rigidity. 2.
In particular, a (k + s, k)-sub-rigid structure is Iso-rigid at order k.

3. The theorem above states that generic lightlike metrics are (3, 1)-sub-rigid.

In the sub-Riemannian case, we have Theorem 3.4. A contact sub-Riemannian metric is (4, 1)-sub-rigid.

Example.

The paradigmatic example of sub-rigid structures presented in [START_REF] Benveniste | Nonexistence of invariant rigid structures and invariant almost rigid structures[END_REF] was that of a degenerate framing. That is, on R n , a system of vector fields x → (X 1 (x), . . . , X n (x)), which are linearly independent everywhere except at an isolated point, say 0. As an example, take n = 1, and the geometric structure being a vector field

X(x) = f (x) ∂ ∂x . A diffeomorphism φ is isometric if φ ′ (x)f (x) = f (φ(x)
). If f does not vanish, then we have a true parallelism, and it is 0-rigid: trivial 0-jet implies trivial 1-jet, say at the point 0 ∈ R; in other words, φ(0) = 0 implies φ ′ (0) = 1.

Assume now that f (0) = 0, then φ is isometric up to order k + 1 at 0, if it satisfies, at the point 0, all the equations obtained by taking derivatives up to order k of the equality: φ ′ (x)f (x) = f (φ(x)). Assume f has a zero of order d at 0, e.g. f (x) = x d , and that jet 1 0 (φ) = 1, i.e. φ(0) = 0 and φ ′ (0) = 1; then we need derivatives of φ up to order d + 2 in order to conclude that jet 2 0 (φ) = 1, i.e. φ ′′ (0) = 0. Therefore, the structure is (d + 2, 1)-sub-rigid. Remarks 3.5.

1. An essentially equivalent example is given in ( [START_REF] Gromov | Lectures on transformation groups: geometry and dynamics[END_REF], §5.11.B) to show weakness of Iso-rigidity in comparison with rigidity.

2. One may think following [START_REF] Benveniste | Nonexistence of invariant rigid structures and invariant almost rigid structures[END_REF] that, as above, there is always a degeneracy phenomenon behind sub-rigidity. One may in particular ask if a sub-rigid structure is rigid on an open dense set? However, the examples of the lightlike structure on the Minkowski lightcone, and the standard contact sub-Riemannian metric on the Heisenberg group, show that sub-rigid structures can be homogeneous. They are in particular nowhere rigid.

PROOF OF THEOREM 3.4

Proof. Let (D, h) be a contact sub-Riemannian structure on M . For computation, it is useful to see (D, h) as an equivalence class of pairs (ω, h) where ω is any contact form defining D. The correspondence (ω, h) → h discussed in §1.0.1 does not depend on the particular choice of ω. We will show that the 1-jet of h is determined by the the 3-jet of (ω, h).

Let X 1 , . . . X 2d a local system of smooth vector fields generating D, where dim M = 2d + 1. The normalized form ω ′ = f ω is defined by

f dω d (X 1 , . . . , X 2n ) = det(h(X i , X j ) ij ) Its Reeb vector field R is defined algebraically by df (X i )ω(R) + f dω(X i , R) = 0, and f ω(R) = 1
If X 0 is a vector field transverse to D, then h(X i , X 0 ) ij , i, j ≥ 0, are given by the h(X i , X j ), i, j > 0, and the coordinates of R in the moving frame {X 0 , . . . X 2d }. In particular the first derivatives of h come from third derivatives of (ω, h).

Let f be a diffeomorphism, f * ω = ω 1 and f * h = h 1 . Thus f * h = h1 . If f is a 4-isometry for (ω, h), then by definition ω and ω 1 (resp. h and h 1 ) coincide up to order 3. It then follows that h and h1 coincide up to order 1, that is f a 2-isometry for h. If f was merely a 4-isometry for (D, h), then, ω will coincide up to order 3 with a multiple gω 1 , which leads to the same conclusion for h.

To prove (4, 2)-sub-rigidity for (D, h), apply the 1-rigidity (say the (2, 1)-subrigidity) of Riemannian metrics. We get here that, if f is a 4-isometry with a trivial 1-jet, then it has a trivial 2-jet.

Example. Endow R 3 with the contact hyperplane field determined by the form ω = dzxdy together the restriction of dx 2 + dy 2 on it. The map f (x, y, z) = (x+ 1 2 z 2 , y -1 2 zx 2 , z + 1 2 yz 2 ) belongs to Iso 3 0 . It has a trivial 1-jet, but a non-trivial 2-jet. Therefore, the structure is not (3, 1)-sub-rigid.

ON THE PROOF OF THEOREM 3.1

We give in what follows hints on the proof of Theorem 3.1. Details, especially for § §5.1.2 and 5.2 will appear in [START_REF] Bekkara | On the local rigidity of degenerate Riemannian metrics[END_REF].

The transversally conformal case.

Let us consider first the transversally conformal case. Locally, M = Q × I, and g (q,r) = c(q, r)h q . An isometry φ has the form φ : (q, r) → (ψ(q, r), δ(q, r)). Since φ preserves the characteristic foliation of g, it acts on the quotient space Q, that is, ψ does not depend on r.

The local isometry equation. The isometric equation is:

c(ψ(q), δ(q, r))φ * h = c(q, r)h That is, φ is conformal, say with a distortion f (i.e. φ * h = f h) which satisfies the cocycle property: c(ψ(q), δ(q, r))f (q) = c(q, r).

INFINITESIMAL CASE. For the sake of simplicity, even for infinitesimal isometries, we will assume that their ψ-part depends only on q (what is a priori true in this case is that the derivatives on ψ with respect to r vanish according to the order of the infinitesimal isometry).

Fix a point, say (q, r) = (0, 0). The fact that φ = (ψ, δ) is isometric of order 1 (and fixes (0, 0)) means exactly that the previous equation is satisfied for (0, 0). So ψ(0) = 0, δ(0, 0) = 0, and ψ is conformal at order 1.

First step: φ ∈ Iso 3 (0,0)(g) and jet 1 (0,0) φ = 1 =⇒ jet 3 0 ψ = 1: -The fact that φ has a trivial 1-jet translates to: D 0 φ = 1, and ∂δ ∂q = 0, and ∂δ ∂r = 1.

-The fact that φ ∈ Iso 2 (0,0) (g) means that the second derivatives of φ at (0, 0) satisfy all the equalities obtained by derivating the previous equation. Here, using that ∂δ ∂q = 0, we observe that we have in fact that ψ ∈ Iso 2 0 (h), that is ψ is not only conformal, but isometric for h. Then, we use 1-rigidity of Riemannian metrics to deduce that jet 2 0 (ψ) = 1. -Now, φ ∈ Iso 3 (0,0) (g) implies in particular that ψ is 3-conformal for h. We then apply Liouville Theorem, that is the 3-rigidity of conformal Riemannian metrics, and deduce that jet 3 0 (ψ) = 1. 5.1.2. Second step: φ ∈ Iso 3 (0,0) and jet 3 0 ψ = 1 =⇒ jet 2 (0,0) δ = jet 2 (0,0) r, i.e. all the second derivatives of δ vanish at (0, 0).

-The equation φ ∈ Iso 3 (0,0) obtained by taking second derivatives of the isometric equation gives relations between jet 2 0 (δ) and jet 3 0 (ψ). Since, we already know that D 0 ψ = 1 and all other derivatives of order ≤ 3 vanish, we get equations relating second derivatives of δ (the first derivatives of δ are known). We then prove that this system of algebraic linear equations (on these derivatives) is determined and that all the second derivatives of δ vanish. 5.2. The general case, generalized conformal structures. When g q,r has a general form rather the split one in the transversally conformal case, we get on the quotient space a kind of generalized conformal structure. This means that at each q ∈ Q, we are giving S q ⊂ Sym 2 * (T q M ), the space of Euclidean scalar products on T q Q, such that S q is the image of a (non-parameterized) curve. The case of Riemannian metrics corresponds to S q reduced to one point, and that of conformal structures to that where all the elements of S q are proportional. (Of course, we assume everything depends smoothly on q). The proof of Theorem 3.1 goes through an adaptation of Liouville theorem to generalized conformal structures, that is a generalized conformal structure is 2rigid.

The second step in the proof of Theorem 3.1 is the same as in the transversally conformal case.

WEAKNESS

We show in what follows how the sub-rigidity is weak in comparison to rigidity. 6.1. Gromov representation theorem for rigid structures. Let G be a Lie group acting on a compact manifold M by preserving an analytic geometric structure σ and a volume form. The Gromov representation theorem concerns the case where σ is rigid and G is a simple Lie group. It states that M tends to look like a quotient G/Λ, where Λ is a co-compact lattice in G. The precise statement is that π 1 (M ) is large, in the sense that it has a representation in some linear group whose the Zariski closure of the image contains a copy of the Lie group G. This result was generalized for actions of lattices in G, by Fisher-Zimmer: Theorem 6.1. [START_REF] Fisher | Geometric lattice actions, entropy and fundamental groups[END_REF] Let Γ be a lattice in a simple Lie group G of rank ≥ 2. Suppose Γ acts on a manifold M analytically by preserving an analytic rigid geometric structure, and ergodically for a volume form. Then, either:

1. Γ acts via a homomorphism in a compact subgroup K ⊂ Diff(M ) (and thus M is a homogeneous space K/C, by ergodicity), or 2. As in Gromov representation theorem, π 1 (M ) admits a homomorphism in some GL N (R) whose Zariski closure contains a subgroup locally isomorphic to the Lie group G.

We will show below that this does not extend to sub-rigid structures.

Extension of Killing fields. One crucial ingredient in the proof on the previous results is that, for rigid geometric structures, local analytic Killing fields extend to the full manifold if it is simply connected (see [START_REF] Amores | Vector fields of a finite type G-structure[END_REF][START_REF] Gromov | Lectures on transformation groups: geometry and dynamics[END_REF][START_REF] Gromov | Rigid transformations groups[END_REF]). More precisely, let M be analytic, simply connected and endowed with an analytic rigid geometric structure σ. Let V be a Killing field of σ defined on an open set U ⊂ M (that is the local flow of V preserves σ). Then, V extends (as an analytic Killing field) to M .

This fact is no longer true for sub-rigid structures. Indeed, let g = x 2d dx 2 be a "singular" Riemannian metric on R. On R -{0}, the metric is regular, and hence flat, it has a Lie algebra of Killing fields of dimension 1. No such Killing field extends at 0. Indeed, as 0 is the unique singular point of g, it will be fixed by any local isometry φ defined on its neighbourhood. One then shows that φ is necessarily ±Id. Indeed, there is a well defined distance d g derived from g. Thus, d g (0, x) = d g (0, φ(x)).

6.2.

No Gromov representation for sub-rigid structures. In the sub-rigid case, we have the following example: Theorem 6.2. The lattice SL 3 (Z) acts analytically and ergodically on a compact simply connected manifold, by preserving an analytic sub-rigid structure and a volume form. More precisely, there exists a holomorphic action of SL 3 (Z + jZ), j = e 2π 3 i , on a compact Calabi-Yau 3-manifold (i.e. a simply connected Kähler manifold with a holomorphic volume form). The action preserves a holomorphic sub-rigid structure, and is ergodic (it is measurably isomorphic to an affine action of a complex torus of dimension 3). Remark 6.3. Observe this is a sub-rigid counter-example for the Fisher-Zimmer version concerning higher rank lattices actions. The original Gromov's theorem deals with actions of Lie groups.

Proof. Before giving the construction, let us discuss somewhat the general question of taking pull-backs of geometric structures.

Pull-Back. Let π : M ′ → M be a differentiable map, with M and M ′ of same dimension n. Assume M is endowed with an H-structure σ (H a subgroup of GL n (R)). If π has no critical points (π a local diffeomorphism) then, one defines straightforwardly π * (σ). Indeed, jet 1 (π) is well defined as a map GL (1) (M ′ ) → GL (1) (M ), and then one composes it with σ : GL (1) (M ) → GL n (R)/H. In contrast, there is generally no mean to define π * (σ) at a critical point. As an example, there is no definition of the inverse image of a vector field on M .

Let us now describe a situation where the definition of the pull-back of an Hstructure is possible as a geometric structure in the Gromov sense, but not as an H-structure. Assume σ is a parallelism x → (e 1 (x), . . . , e n (x)) on M . One defines a geometric structure σ ′ : GL (1) (M ′ ) → Mat n (R) by: u = (u 1 , . . . , u n ) ∈ GL (1) y (M ′ ) → σ ′ (u) = (a ij (u)) ∈ Mat n (R) where D y π(u i ) = Σa ij (u)e j (π(y)). In other words, σ ′ (u) is the matrix of the derivative D y π with respect to the bases (u 1 , . . . , u n ) and (e 1 (π(y)), . . . , e n (π(y))) of T y M ′ and T π(y) M , respectively.

Case of the affine flat connection. Another situation which serves in the proof of our theorem is that of the usual affine structure on R n . This an H-structure of order 2, i.e. a map σ : GL (2)

(R n ) → GL (2) 0 (R n ), where GL (2) 0 (R n ) is the set of invert- ible jets in jet 2 0 (R n , R n ) (the space of jets (R n , 0) → R n ), i.e. the inverse image of GL n (R) under the projection jet 2 0,0 (R n , R n ) → GL n (R n ) ⊂ jet 1 0,0 (R n , R n ) (the space of jets (R n , 0) → (R n , 0)). The usual affine connection on R n is obtained from the projection GL (2) (R n ) = R n × GL (2) 0 (R n ) → GL (2) 0 (R n ). Now the point is that σ extends as a map σ : jet 2 0 (R n , R n ) → jet 2 0,0 ((R n , R n ). The smooth map π always induces a map jet 2 (π) : GL (2) (M ′ ) → jet 2 0 (R n , R n ). We define π * (σ) as σ • jet 2 (π).
If the degeneracy of π is bounded, that is there exists k such that jet k x (π) = 0, for any x, then π * (σ) is sub-rigid.

The construction is natural, and thus, if a group Γ acts on M ′ and R n equivariantly with respect to π, and if the action on R n is affine, then, Γ preserves the pull-back of the geometric structure on M ′ .

All this applies identically to the torus T n = R n /Z n , since we have the same trivialization of the jet bundle GL (2) (T n ).

Actions. Consider Γ a subgroup of SL n (Z) and let it act as usually on T n . Blowup a finite orbit of Γ (e.g. a rational point), and get a manifold M ′ with a Γ-action. It was proved by Katok-Lewis [START_REF] Katok | Global rigidity results for lattice actions on tori and new examples of volume-preserving actions[END_REF] that this action is volume preserving, and by Beneviste-Fisher [START_REF] Benveniste | Nonexistence of invariant rigid structures and invariant almost rigid structures[END_REF] that it preserves a sub-rigid structure, but can not preserve a rigid one.

Orbifolds. Here, we assume that there is a finite index subgroup F ⊂ SL n (Z) commuting with Γ. We then consider the orbifold M 0 = T n /F . It inherits a Γ-invariant natural flat affine connection in an orbifold sense.

The next step is to desingularize M 0 in order to get a (regular) manifold M ′ with a Γ-action equivariant with respect to a projection M ′ → M 0 .

An example. Take F to be the group isomorphic to Z/2Z generated by the involution I : x → -x. If n = 2, the quotient around fixed points is just a cone with opening angle π. It follows that the so obtained orbifold is a topological surface. It is in fact a topological sphere, with exactly 4 conic singularities on which Γ = SL 2 (Z) acts continuously by preserving a continuous volume form. The singularities can be solved to give the usual differentiable structure on the sphere, but this can not be done Γ-equivariantly.

Complex case. A higher dimensional generalization is possible, but in a complex framework. So, we start with a complex torus A = C n /Λ (of complex dimension n). We consider M 0 = A/F , where F is the same previous group generated by the involution I. If n ≥ 2, M 0 is no longer a topological manifold, since the fixed points of I are not conical. We then start blowing-up A on I-fixed points, in a complex way, and get M 1 . We have a projection p 1 : M 1 → A with singular fibers isomorphic to CP n-1 over I-fixed points. Now, F acts naturally on M 1 with conical singularities, and hence M 2 = M 1 /F is a topological manifold. The resolution of singularities yields a complex structure on M 2 with a natural Γ-holomorphic action.

Our case. For n = 2, we get a Kummer surface, a special case of K3 surfaces. Observe that the volume form dv = dz 1 ∧dz 2 is F -invariant and hence well defined on M 2 . However, even if the form p * 1 (dv) is singular along the exceptional fibers, it gives rise to a true regular holomorphic volume form on M 2 .

In order to have a similar construction in dimension 3, we replace F by the group generated by the rotation J : z → jz where j = e 2π 3 i on C 3 (see [START_REF] Cantat | Holomorphic actions, Kummer examples, and Zimmer program[END_REF], §7.6). It preserves the volume form, and therefore, we get on the corresponding M 2 a holomorphic volume form.

Regarding the Γ-action, we take, Γ = SL 3 (Z + jZ) and Λ = (Z + jZ) 3 ⊂ C 3 . Thus, Γ is a lattice in SL 3 (C), it preserves Λ and commutes with J.

As in the cases n = 1, 2, one can prove directly that M 2 is simply connected. Another idea is to use the fact that M 2 has holomorphic volume form to deduce it has a vanishing first Chern class. Then, apply Yau's theorem to get a Kähler Ricci flat metric on it. But, for such manifolds, up to a finite cover, there is a de Rham decomposition into a product of a flat torus, and (compact) simply connected manifolds (hyper-Kähler and Calabi-Yau, see [START_REF] Beauville | Variétés kähleriennes dont la première classe de Chern est nulle[END_REF]). Thus, it suffices to verify that M 2 has no torus (of dimension 1, 2 or 3) as a factor, to prove that it has a finite fundamental group. Remark 6.4. One can use general theory of rigid transformation groups to see that the latter action can not preserve a (real) analytic rigid geometric structure. Indeed, by [START_REF] Gromov | Lectures on transformation groups: geometry and dynamics[END_REF][START_REF] Gromov | Rigid transformations groups[END_REF], the isometry group of a unimodular analytic rigid structure on a simply connected manifold have a finite number of connected components. This means that up to a finite index, the Γ-action extends to an action of a Lie group, which can be easily seen to be impossible.

Supported in part by the ANR Geodycos of the ENS-Lyon and the project RIAMI of the CIMPA.


Frédéric Ablitzer

Jean-Pierre Dalmont

Nicolas Dauchez

Static model of a violin bow: influence of camber and hair tension on mechanical behavior

Keywords: numbers: 43, 75, De

Experienced bow makers empirically know the influence of wood, tapering and camber on the playing and tonal qualities of a bow. However, the way each parameter affects the bow mechanical behavior is not clearly established. An in-plane finite element model is developed to highlight the link between the adjustable design parameters and the mechanical behavior of a bow. This model takes into account geometric nonlinearity as well as compliance of the hair. Its validity is discussed from measurements on a bow. Results from simulations are compared to experimental results from previous studies. The consequences of adjusting hair tension and camber are then investigated.

I. INTRODUCTION

In spite of an extensive scientific literature about the violin [START_REF] Cremer | The physics of the violin[END_REF][START_REF]Research papers in violin acoustics[END_REF] , only few studies are exclusively devoted to the bow. However, experienced players attach almost as much importance to the choice of a bow as to the choice of a violin. They generally asses the quality of a bow in terms of playing and tonal qualities. Playing qualities refer to the control of bow in playing, tonal qualities to the influence on the tone. The question of a link between physical properties and quality of bows was addressed by several authors [START_REF] Schumacher | Some aspects of the bow[END_REF][START_REF] Pickering | Physical characteristics of violin bows[END_REF][START_REF] Askenfelt | A look at violin bows[END_REF][START_REF] Askenfelt | Observations on the violin bow and the interaction with the string[END_REF][START_REF] Caussé | Study of violin bow quality[END_REF][START_REF] Dauchez | Qualité des archets de violon et instabilité de type flambement" (Quality of violin bows and buckling instability)[END_REF][START_REF] Ravina | Experimental modal analysis of bows[END_REF][START_REF] Rossing | The science of string instruments, chapter "Bows, Strings, and Bowing[END_REF] . Static properties such as total mass, mass distribution, and bending stiffness, have generally been assumed to affect the playing qualities, while dynamic properties such as hair and stick modes would relate to the tonal qualities [START_REF] Askenfelt | A look at violin bows[END_REF][START_REF] Askenfelt | Observations on the violin bow and the interaction with the string[END_REF][START_REF] Guettler | Some aspects of bow resonances -conditions for spectral influence on the bowed string[END_REF] . In most studies, some static and dynamic properties were measured on different bows with various degrees of quality. Despite the seeming simplicity of the bow compared to the violin, these studies showed that it is not yet possible to predict the quality of a bow by measuring a set of physical properties. However, some authors pointed out the plausible existence of an acceptable range for certain characteristics, out of which the bow would be considered as not suitable for playing [START_REF] Pickering | Physical characteristics of violin bows[END_REF][START_REF] Askenfelt | Observations on the violin bow and the interaction with the string[END_REF] .

When manufacturing a bow stick, the bow maker works with three main parameters: wood, tapering and camber [START_REF] Rolland | Sticking point[END_REF] . Regarding wood, high quality modern bows are made of pernambuco (Caesalpinia Echinata), a Brazilian wood with high specific Young's modulus and low damping. However, this species has become rare and expensive. Tapering denotes the gradually decreasing thickness along the stick. In general, the maker adjusts the taper of the bow in order to reach the desired total mass, for a given wood density. Camber is the concave curvature of the stick without hair tension. It is adjusted by heating a short portion of the stick over a flame and bending it until it cools down. Once a bow is finished, the only possible adjustment by the player is the hair tension.

This paper aims at highlighting the link between the adjustable design parameters and the mechanical behavior of a bow. Emphasis is put on camber and hair tension, since both a) Electronic address: frederic.ablitzer.etu@univ-lemans.fr can be adjusted on a finished bow. For this purpose, an in-plane finite element model is developed and validated from measurements on bow. It is then used to investigate the effects of hair tension and camber on the mechanical behavior of a standard bow.

II. MODEL OF THE ASSEMBLED BOW

In previous work, various ways of modeling a bow have been proposed. Wegst and Ashby [START_REF] Wegst | Alternative woods for violin bows[END_REF] made simple calculation from beam theory, considering a stick with a circular cross-section of constant radius, to determine a typical range of Young's modulus required for violin bows. Pitteroff [START_REF] Pitteroff | Contact mechanics of the bowed string[END_REF] proposed an analytical model aimed at describing the static behavior of the assembled bow, i.e. stick and hair, in the perpendicular plane. Despite rather rough approximations, the model agreed well with experimentally observed behavior for moderate bow forces. In his model, the stick was represented by its stiffness at the end, which can be written analytically in the case of a simplified geometry. To take into account a more realistic geometry, other authors employed numerical models. Carlsson and Tinnsten [START_REF] Carlsson | Geometrical compensation for varying material properties in bows by the use of numerical optimization[END_REF] used a finite element model coupled to an optimization algorithm to calculate the change in diameter along the stick needed to recover some of the static and dynamic properties after changing density and Young's modulus. The properties of the tightened bow were not investigated in this study.

Caussé et al. [START_REF] Caussé | Study of violin bow quality[END_REF] made simulations based on the finite element model of an assembled bow to highlight mode shapes of stick and hair and to evaluate the influence of hair tension on corresponding eigenfrequencies. In their study, the stress field due to hair tension was taken into account to determine the eigenmodes. This was done in a similar way as Mamou-Mani et al. [START_REF] Mamou-Mani | Numerical simulation of a piano soundboard under downbearing[END_REF] studied the effect of downbearing on piano soundboard eigenfrequencies, but the geometrically nonlinear nature of the problem was not discussed in the case of the bow.

However, the change in geometry during the loading of the bow, e.g. when tightening the hair, is strong enough to cause geometric nonlinearity, even under hypothesis of small deformation and linear behavior of the material, as pointed out by Dauchez et al. [START_REF] Dauchez | Qualité des archets de violon et instabilité de type flambement" (Quality of violin bows and buckling instability)[END_REF] . In the present paper, a finite-element model of the bow including the hair and taking into account geometric nonlinearity is now developed. As a starting point, only the in-plane static behavior of the bow is considered.

A. General description

The bow as we know it today is the result of a progressive evolution through centuries [START_REF] Ablitzer | Mechanics of the violin bow: relationship between evolution and musical repertoire[END_REF][START_REF] Poidevin | Premiers archets à travers l'iconographie" (Early bow in iconographic sources)[END_REF] . Renaissance bows had a short thick stick with convex curvature. As the musical style changed, the stick got longer and thinner, while its curvature was progressively changed from convex to concave in order to withstand the hair tension without increasing the mass too much. At the beginning of the 19th century, the french bow maker François-Xavier Tourte brought the last significant improvements to the design of bows. Today he is considered the father of the modern bow.

A modern violin bow is represented in Figure 1. The stick is traditionnally made of wood. Its main function is to support a 65 cm hair ribbon under tension. The hair is attached to the head of the stick at one end, and to the frog at the other. The hair enters the head through a thin plate, traditionally made of ivory, which covers the underside of the head. The position of the frog is adjusted by turning the button, which operates on a screw mechanism located inside the stick. This allows the player to tighten the hair to suitable tension for playing. The right tension is essentially determined by the transverse compliance of the hair. If too loose, the hair tends to touch the stick in playing. If too tight, more effort is needed to make the string vibrate, according to players and bow makers [START_REF] Grütter | A bow on the couch[END_REF][START_REF] Rolland | The playing parts of the bow: focusing on the stick[END_REF] . On early bows, the frog was simply wedged between stick and hair. The resulting hair tension was determined by the height of the frog and the length of the loose hair, both fixed by the bow maker.

The model presented in this paper allows one to simulate, first, the tightening of the bow from its initial state without hair tension [Fig. 2 The model just described is a simplified representation of actual playing conditions. In reality, the bow is held in a finger grip which allows the pivoting around an axis located somewhere near the cut up in the frog, at the position of the thumb. The normal force at the contact point between the hair and the string is controlled by applying a moment with the index finger on top of the stick, with the thumb acting as a support. Since the combined action of index and thumb fingers takes place near the frog, on a thick part of the stick, it is assumed that this modeling makes little difference with the actual static behavior of the bow in playing.

The model takes into account the geometric nonlinearity of the bow as well as the compliance of the hair. It is based on a finite element formulation coupled with an iterative procedure.

B. Finite element model of the bow

In order to take into account the geometric nonlinearity, the model is based on the corotational formulation for beams given by Crisfield [START_REF] Crisfield | Non-linear finite element analysis of solids and structures[END_REF] . The corotational formulation allows the global displacements and rotations of a structure to be arbitrarily large, although the local strain are assumed to remain small (Fig. 3). A local reference frame is attached to each element and continuously rotates and translates with it. With respect to this local frame a small-strain, small-displacement relationship is applied. Since a bow stick is slender, this relationship can be derived from the Euler-Bernoulli beam theory. For each beam element, the axial force N is related to the local axial displacement u l by the relation

N = E A u l l 0 , ( 1
)
where E is the element Young's modulus, A the element area and l 0 the element initial length. The internal bending moments M1 and M2 are related to the local nodal rotations θ l1 and θ l2 by the relation

     M1 M2      = E I l 0    4 2 2 4         θ l1 θ l2      , ( 2
)
where I is the element second moment of inertia.

The original formulation developed by Criesfield leads to the following relationship between variation of global internal forces δq i and variation of global displacement δp,

δq i = (K t1 + K tσ ) δp , (3)
where K t1 is the standard tangent stiffness matrix, K tσ the geometric stiffness matrix. K t1 and K tσ are displacement-dependent matrices.

In order to allow a sufficiently fine representation of tapering, the bow stick is discretized into 20 beam elements. The head is represented by a single element with diameter 10 mm, so as to make it sufficiently stiff. After assembling, the problem to solve takes the form:

K(u) u = f(u) , ( 4
)
where u is the displacement vector, K(u) the assembled stiffness matrix and f(u) the external force vector. The only external force in f(u) is the force T exerted by the hair on the stick at the tip [see Fig. 2(c)], which varies in orientation and amplitude in function of the displacement of the tip. The variation in orientation is due to the rotation of the hair coordinate system. The variation in amplitude is due to the increase in hair tension ∆T = T -T 0 as the tip moves away from its position at T 0 . Since K(u) and f(u) are displacement-dependent, an iterative procedure is necessary to solve Eq. (4). Moreover, an additional stiffness term for the last element, i.e. the head, has to be introduced to take into account the variation in force orientation and amplitude between two successive configurations. The additionnal stiffness term is expressed in a similar way as when follower forces are involved [START_REF] Argyris | Dynamics of structures[END_REF] .

For the general case where the solution of the tightened and loaded bow is seeked, the numerical procedure achieves good convergence when decomposing the computation into two global load steps:

1. tightening of the bow, i.e. T x = T 0 and T y = 0, 2. loading of the hair, i.e. T x = T 0 + ∆T x and T y = 0, where T x and T y denote the components of T. For both steps, the Newton-Raphson algorithm with several load increments is applied to reach the equilibrium.

Regarding the first step, a backward displacement of the frog relative to the stick (typically 3-3.5 mm) is normally needed to tighten a bow. This displacement counterbalances the stretching of the hair (1 mm) and the decreasing distance between the frog and the tip (2-2.5 mm) due to straightening of the stick and rotation of the head, as reported by Pitteroff [START_REF] Pitteroff | Contact mechanics of the bowed string[END_REF] . In the model, neither the stretching of the hair at this step, nor the displacement of the frog is considered. Therefore, the distance between the frog and the tip decreases when tightening the modeled bow, whereas it slightly increases on an actual bow, because of the elongation of the hair. Although the increase in bow length is not fully negligible, it does not affect the overall behavior examined in the second step. Once the bow is tightened, the distance between frog and tip defines the hair length L 0 corresponding to tension T 0 . In order to compute the value of T x and T y to apply in the second load step, a model of the hair is now defined.

C. Model of the hair

The ribbon of hair is assumed to behave like an equivalent single hair, represented in Figure 4. When an external force F is applied at relative abscissa γ (0 < γ < 1), the hair on either side is assumed to extend proportionally to the increase in tension ∆T = T -T 0 , where T is supposed to be the same on either side, i.e.

T = ||T|| = ||T ′ || = T 2 x + T 2 y .
This implies that the force F has a small tangential component F x of unknown value (except for γ = 0.5 where it is null).

It has to be noted that the distribution of forces in the x and y directions, respectively, is slightly different under normal playing conditions. In this situation, the tangential component of F corresponds to the friction force, i.e. F x = µ F y , where the coefficient of friction µ varies within each cycle of Helmholtz motion [START_REF] Schumacher | Measurements of some parameters of bowing[END_REF][START_REF] Woodhouse | Reconstruction of bowing point friction force in a bowed string[END_REF] . As the force F is fully determined, the tension may differ on either side of bowing point. Cremer 1 reported a value of 0.24 for the average coefficient of friction. With a typical bow force of 1 N, the corresponding average friction force is fairly small compared to hair tension. Thus, the difference between ||T|| and ||T ′ || is likely to remain small, which gives support to the hypothesis made here.

L1 L 2 L h F x F -T x -T y F y δ h ÖÓ γL 0 L 0 Ø Ô ´ µ ´ µ T 0 -T 0 -T T ′ FIG. 4.
Model of the hair. Tightened hair (a), deflection under loading (b).

The lengths of both sides are:

L 1 = γ L 0 (1 + c h ∆T ) L 2 = (1 -γ) L 0 (1 + c h ∆T ) , (5)
where c h is the compliance per unit length of the equivalent single hair. Considering a ribbon of n h hairs having identical and homogeneous Young's modulus E h and diameter d h , an estimation of c h can be obtained from:

c h = 4 n h E h π d 2 h . (6)
Typical values for a violin bow are n h = 160-190, E h = 4-7 GPa and d h = 0.2 ± 0.05 mm, as reported by Askenfelt [START_REF] Askenfelt | Observations on the violin bow and the interaction with the string[END_REF] . He also measured the stiffness constant of a complete ribbon and found it to reach 30 N.mm -1 under a nominal tension of 60 N. With a length L h = 650 mm, the corresponding compliance per unit length is

c h = 5.1 × 10 -5 N -1 .
The hair deflection at the loading point is

δ h = L 2 T y T . (7)
The distance between frog and tip is

L h = L 2 1 -δ 2 h + L 2 2 -δ 2 h . (8)
T y = γ F y . (9)
The tangential component cannot be expressed by a similar equation. Therefore, it is found iteratively, starting from the initial guess T x = T 0 . At each computation step of the Newton-Raphson algorithm, the value of L h is computed from Eq. ( 8) and compared to the distance between frog and tip in the current configuration. As long as the two values differ, the value of T x is corrected for the next iteration.

D. Validation of the model

In this section, the ability of the finite element model to reproduce the deformed shape of an actual bow is examined. A wooden student violin bow was tightened from zero to high playing tension, by steps of two turns of the button. At each level, the shape of the stick was determined from a picture, by means of image processing [Fig. 5(a-c)]. The resolution in the measurement was about 0.3 mm. To keep the imaging conditions as identical as possible for all pictures, the position of both the bow and the camera were held constant. The diameter along the stick was measured with a digital caliper. The relative incertitude on the diameter was less than 3%. The Young's modulus was then determined from two measurements of the dynamic response of the stick simply supported at its ends and loaded by a mass at its center (200 g and 400 g), by means of the measuring platform Lutherie tools [START_REF] Gautier | Lutherie tools : projet collaboratif entre ateliers de lutherie et laboratoires" (Lutherie tools: a project between instrument makers and laboratories)[END_REF] . A value of 36 GPa was found.

Then, simulations were performed starting from the initial geometry. For each level of hair tension, the value of T 0 giving the best least-squares fit between measured shape and model output was determined. The maximum difference between the experimental and numerical results on the y-coordinate of the neutral axis was less than 0.2 mm. It should be noted that this procedure can be considered as an indirect measurement of hair tension, provided that the Young's modulus is known. With the bow used in this experiment, values of T 0 = 20.1 N, 39.2 N, 54.9 N and 67.5 N (resp. at 2, 4, 6 and 8 turns of the button) were obtained. The levels of hair tension corresponding to 4 and 8 turns of the button were considered as the lower and upper limits for playing, respectively. The optimal playing tension was found at 6 turns (T 0 = 54.9 N). Although a typical value of 60 N is commonly refered to as "normal" playing tension by authors [START_REF] Pitteroff | Contact mechanics of the bowed string[END_REF][START_REF] Askenfelt | Observations on the dynamic properties of violin bows[END_REF][START_REF] Demoucron | Measuring bow force in bowed string performance: Theory and implementation of a bow force sensor[END_REF] , the optimal value for each bow probably depends on the stiffness of the stick and camber as well. Figure 5(d) shows the nonlinear increase in minimum hair-stick distance with hair tension.

III. RESULTS

In this section, we first describe the geometry of a "standard bow" on which simulations are performed. Then, general results from simulations are presented and discussed with respect to experimental observations by other authors. Finally, the influence of initial hair tension and camber on transverse bow compliance is examined.

A. Definition of a standard bow

The geometry of the modeled stick, which is basically defined by camber and tapering, can be taken from measurements on a representative modern violin bow. In this study, however, we define the geometry of a "standard bow" from criteria found in the literature. Similarly, the Young's modulus E is fixed at 25 GPa, which is a typical value for pernambuco [START_REF] Carlsson | Geometrical compensation for varying material properties in bows by the use of numerical optimization[END_REF][START_REF] Matsunaga | Physical and mechanical properties required for violin bow materials[END_REF][START_REF] Schimleck | Characterization of the wood quality of pernambuco (Caesalpinia echinata lam) by measurements of density, extractives content, microfibril angle, stiffness, color, and NIR spectroscopy[END_REF] . Measurements of a set of bows in professional use today would be needed to check whether this "standard bow" is representative of most bows available on the market.

Tapering

The tapered profile is derived from a formula given by Vuillaume. This violin maker of the 19th century measured a great number of Tourte bows and found the diameter to decrease logarithmically along the stick [START_REF] Fétis | Notice of Anthony Stradivari, the celebrated violin-maker, known by the name of Stradivarius, chapter "Experimental determination of the form of Tourte's bows[END_REF] . The corresponding curve is considered as a reference in some studies [START_REF] Pickering | Physical characteristics of violin bows[END_REF][START_REF] Hopfner | Zur Entwicklung des Streichbogens anhand von Objekten der Sammlung alter Musikinstrumente, Wien[END_REF] . The original formula applies to abscissa between 110 mm and 700 mm from the stick origin and corresponds to a decrease in diameter from 8.6 mm to 5.3 mm, the first 110 mm of the stick having constant diameter of 8.6 mm. Since the origin of the modeled bow is the front end of the frog, the formula is adapted so that it applies to abscissa between 0 mm and 650 mm, neglecting the small portion of constant diameter [Fig. 6(a)]. Thus, the diameter d at abscissa x is given by:

d(x) = d 0 1 + ε ln x ∞ -x x ∞ , (10)
where d 0 = 8.77 mm, x ∞ = 825 mm and ε = 0.255. Interestingly, the second moment of inertia I(x) following from Eq. ( 10) decreases almost linearly with abscissa for this set of parameters. It can be approximated by

I(x) ≈ I 0 x -x 0 x 0 , (11)
where I 0 = 293 mm 4 and x 0 = 752 mm, with 3% maximum relative difference with that deduced from Eq. (10). Since this observation has no obvious physical interpretation, one may wonder whether this particular tapered profile is an optimum empirically found by Tourte.

Camber

The concept of camber actually includes two aspects: distribution of camber, which can be seen as shape of the stick; amount of camber, which represents how strong the stick is bent. The distribution of camber is here defined so that the stick becomes straight under a certain tension, which is a criteria commonly recognized by bow makers: "The match between wood strength and camber can be tested by tightening the bow until the stick is straight (...) If the stick is really straight, camber and wood strength are properly matched" (Grütter [START_REF] Grütter | A bow on the couch[END_REF] ); "One reference: the axis of the stick should be perfectly parallel to the line of hair at maximum tension of the bow" (Rolland 20 ). The maximum tension, i.e. Following this reasoning and considering the approximation of Eq. ( 11), a simple analytical expression giving the initial shape of the stick y(x) can be provided by integrating

d 2 y dx 2 = h T max 0 E I(x)
(with y(0) = h and y ′ (0) = 0), which yields

y(x) -h = h T max 0 x 0 2 E I 0 x + (1 -x) ln(1 -x) , (12)
where x = x x 0 . The multiplying factor in Eq. ( 12), in terms of E, I 0 , and T max 0 , is considered to be the amount of camber. The expression within the parentheses, dependent on x, gives the generic shape of the stick without hair tension (distribution of camber).

is the input parameter that determines the amount of camber of the modeled stick. Althought it corresponds to a physical quantity, a more convenient way of quantifying the amount of camber is the minimal distance between loosened hair and stick, denoted by κ. The smaller this distance, the more camber the stick has. It is generally agreed among bow makers that this distance should be between 0 mm and 2 mm and in any case not negative. The case where κ = 0 mm is called "full camber".

In this study, four different amounts of camber are investigated. The value of T max 0 needed to obtain each chosen value of κ is determined by means of the finite element model. The corresponding values of κ and T max 0 are listed in Table I.

Comment on the relationship between tapering and camber

In playing conditions, the bow is never tightened up to complete straightening of the stick. Therefore, the bending moment resulting from hair tension is a function of abscissa x,

M 0 (x) = T 0 a(x)
, where a denotes the distance between hair and neutral axis of the stick. It is maximum at the tip and at the frog (both head and frog have standard height on modern bows) and minimum at the lowest point of the stick, generally near the middle. Moreover, the stiffness of the stick is determined by the quantity E I, where Young's modulus E is supposed to be homogeneous along the stick, contrary to second moment of inertia I which depends on tapering. For a circular section, I = π d 4 64 . Thus, the stick is far more compliant at the tip than at the frog. The local bending radius ρ that counteracts the initial curvature of the stick is given by:

1 ρ = M 0 E I . ( 13
)
From Eq. ( 13) it is clearly apparent that the stick will bend more near the tip, where the bending moment is maximum and the stiffness minimum. For this reason, camber is particularly strong in the last portion of the stick. More generally, the role of distribution of camber is to compensate for the decrease in diameter along the stick. In addition, the more cambered the stick, the less it will straighten under the same tension. Since tapering is mostly adjusted so that the stick reaches a given mass, the bow maker has to adjust camber accordingly, in order to get the right shape on the tightened bow. A physical model can be useful to determine the proper camber for a given tapering, as pointed out by a recent study [START_REF] Graebner | Optimizing the taper-camber relationship in bows for string instruments. (a)[END_REF] .

B. Variation in hair tension under transverse loading

In previous studies on violin bows, the influence of hair tension on different properties of the bow, e.g. bending stiffness along the stick [START_REF] Askenfelt | Observations on the violin bow and the interaction with the string[END_REF] , stick mode frequencies [START_REF] Caussé | Study of violin bow quality[END_REF] , bouncing frequency [START_REF] Askenfelt | Observations on the dynamic properties of violin bows[END_REF] , has been examined. In most studies, the hair tension has been assumed not to vary significantly from the initial value fixed by the player before playing. However, it has been experimentally shown by Demoucron et al. [START_REF] Demoucron | Measuring bow force in bowed string performance: Theory and implementation of a bow force sensor[END_REF] that the hair tension does vary when the bow is loaded by a normal force. The tension variation was found to be almost linear with respect to relative abscissa γ and normal force F y , leading to a simple empirical relation,

T = T 0 + α T γ F y , (14)
where α T is an experimentally determined coefficient.

Figure 7 shows the simulated hair tension variation when the bow is loaded by a normal force F y of increasing value at relative abscissa γ. The tension increases almost linearly with relative abscissa and normal force, which is in agreement with the observations of Demoucron. For a normal force of 1.5 N at the middle and 1.0 N at the tip, which are moderate values in violin playing, the hair tension increases by 20% and 25%, respectively. For a normal force of 1.5 N at the tip, it increases by 40%. As a comparison, Demoucron measured a variation of 25% for the same load case. This discrepancy could be due to the higher stiffness of the bow he used for the experiment (stiffness at the tip K s = 91 N.m -1 ) compared to the modeled bow (K s = 72 N.m -1 ), since the increase in hair tension is essentially due to the displacement of the tip under loading.

C. Distribution of transverse compliance along the bow

When applying a normal force to the tightened bow hair, the observed deflection at the loading point is the result of two effects: deflection of the hair and bending of the stick.

The force transmitted by the hair to the tip makes the stick bend. This results in a second displacement term, in addition to the hair deflection.

Pitteroff [START_REF] Pitteroff | Contact mechanics of the bowed string[END_REF] considered the two effects separately to give an analytical relationship between force and deflection. The contribution of the hair was the deflection δ h of a string with fixed length L h and tension T 0 under normal force F y at relative abscissa γ. The contribution of the stick was deduced from the deflection δ s of a cantilever beam under normal force γF y at its free end. The stick was assumed to be straight and have constant bending stiffness.

However, the formula remains valid when considering a more realistic geometry, simply by introducing the equivalent stiffness constant at the tip K s [START_REF] Demoucron | Measuring bow force in bowed string performance: Theory and implementation of a bow force sensor[END_REF] . The total deflection at the loading point was expressed as the sum of the two contributions:

δ = δ h + δ s = γ(1 -γ)L h T 0 F y + γ 2 K s F y . ( 15
)
When compared to experimental results, this relationship matched well with the behavior of an actual bow in its lowest two thirds. However, a significant difference between experimental and theoretical deflection was observed at the tip.

The model developed in the present article removes the hypothesis that the effects of hair and stick are uncoupled. Yet, it is possible to estimate how hair and stick contribute to the total deflection. The hair deflection is given by Eq. ( 7). The total deflection is deduced from coordinates of the loading point before and after the force is applied. The contribution of the stick is then given by the difference between total deflection and hair deflection.

Figure 8 shows the simulated total deflection at the loading point of the standard bow loaded by a 1 N normal force plotted against relative abscissa of the force. Hair and stick contributions are also plotted individually. The total deflection is null at the frog and maximum at the tip. The stick deflection continuously increases along the bow, reaching a maximum at the tip. On the contrary, the shape of the hair contribution is similar to a parabola, though it is slightly asymmetrical: the maximum value is reached just before the midpoint of the bow. This asymmetry is due to the rise in hair tension with relative abscissa.

As a comparison, it is possible to estimate the parameters T 0 and K s used by Pitteroff by fitting the total deflection with a second order polynomial of the same form as Eq. ( 15)

and identifying the coefficients. The third parameter, L h , is known from the finite element model. From this method, the estimated value of hair tension is found to be 24% higher than its actual value, T 0 = 45.0 N. The same operation was done for three other values of F y . The estimated values for T 0 and K s , as well as the maximum relative difference in deflection between numerical results and the second order polynomial, are given in Table II.

Regarding the hair tension, the deviation from actual value increases with force F y . Similarly, the estimated value of K s varies significantly with F y . This shows that Eq. ( 15) does not Moreover, Pitteroff observed that the transverse compliance of the bow varies with the normal force. In the lower half, the measured compliance was lower with a higher force (5 N compared to 1 N). On the contrary, the compliance near the tip increased with the force.

The nonlinearity of transverse compliance close to the tip was also reported by Askenfelt [START_REF] Askenfelt | Observations on the violin bow and the interaction with the string[END_REF] .

Figure 9 shows the simulated deflection of the bow as a function of normal force at three different loading points. The normal force F y is limited to the range between 0 N and 1.5 N.

Although the bow force generally remains low when playing near the tip, it may well reach values around 1.5 N in this part of the bow [START_REF] Askenfelt | Measurement of bow motion and bow force in violin playing[END_REF] . A comparison with a linear case, indicated by dotted lines in Figure 9 can be made. As seen, the simulated transverse compliance is nonlinear. Close to the frog as well as in the middle, the compliance tends to diminish as the force increases. This is due to the increase in hair tension with force. Near the tip, the compliance increases with the force.

D. Influence of initial hair tension on transverse compliance

Before playing, the player adjusts the initial hair tension by turning the button. As the tension increases, the stick straightens progressivly, which increases the minimum distance between hair and stick. The evolution of minimum hair-stick distance with tension for four settings of camber (4 mm, 2 mm , 0 mm and -2 mm) is plotted in Figure 10. The increase in hair-stick distance with tension is far from linear. This is due to the fact that the straightening of the stick, which increases the hair-stick distance, is the consequence of a bending moment which itself depends on the hair-stick distance.

To evaluate the influence of initial hair tension on the transverse compliance of the bow, simulations were performed for four values of T 0 on the standard bow with full camber.

The values of T 0 (40.5 N, 45.2 N, 49.0 N, and 52.1 N) correspond to four plausible values of hair-stick distance, from 8 mm to 14 mm, denoted by numbers (0) to (3) in Figure 10.

Because of nonlinearity, the transverse compliance depends on normal force. It is defined by the slope of the force-deflection curve (see Fig. 9).

Figure 11 shows the transverse compliance along the bow with full camber corresponding to the four chosen values of initial hair tension. For each setting, the two curves indicate the compliance for small forces (just above 0 N) and high forces (around 1.5 N). The compliance for small forces slightly decreases with initial hair tension. Increasing T 0 from 40.5 N to 52.1 N (29%) lowers the compliance for small forces at the middle of the bow by 18% and by 13% at the tip. Moreover, the nonlinearity is stronger for a low hair tension.

E. Influence of camber on transverse compliance

When varying camber in the model, it is necessary to change the initial hair tension accordingly so that the minimum hair-stick distance remains in a range which is suitable for playing. For each setting of camber in Figure 10, the tension at which a 10 mm hair-stick distance is reached is denoted by T play 0 and called "playing tension". Figure 10 shows the four settings of camber, denoted by letters (G) to (J). Table I gives the corresponding values of κ, T play 0 and tension for straight stick T max 0 . None of the different settings of camber and hair tension led to contact between hair and stick within the chosen ranges of relative abscissa and normal force.

The first noticeable effect of camber is a change in playing tension. Increasing camber from κ = 4 mm up to κ = 0 mm, for example, increases the hair tension by 66% for the same hair-stick distance. It is noteworthy that not only the minimum hair-stick distance is preserved, but also the hair-stick distance throughout the bow. Thus, the bow maker can increase the playing tension of a bow without any consequence on its shape once tightened.

A second effect of camber is a change in transverse compliance (see Fig. 12). As camber increases, the compliance at small forces decreases in the middle (-22% from κ = 4 mm to 0 mm), which is due to a higher playing tension. At the tip, the compliance at small forces increases with camber (+10% from κ = 4 mm to 0 mm), contrary to what was observed by increasing the hair tension at given camber (Fig. 11). This effect is due to prebending of the stick. Furthermore, if the bow is not cambered very much (G), the nonlinearity is fairly strong around the middle: the higher the force, the lower the compliance. Close to the tip, the reverse trend is observable though moderate. As camber increases, the nonlinearity tends to decrease around the middle of the bow, whereas it increases near the tip. At full camber (I), the compliance in the lower two-thirds is almost the same for small and high forces, whereas the compliance near the tip significantly increases with force. Moreover, the range in length for which the compliance increases with force becomes larger as camber increases. If the bow has an unrealistically large amount of camber (negative κ, point J), the compliance for high forces strongly increases in the last third.

IV. CONCLUSION

A finite element model of the assembled bow accounting for the geometrical nonlinearity of both stick and hair has been presented. The model allows to reproduce the in-plane nonlinear static behavior that is experimentally observed when the bow is loaded (Fig. 2).

The influence of the adjusting parameters (camber, hair tension) on the mechanical behavior of the bow has been enlightened. Changing the hair tension does not affect the overall profile of the transverse compliance of the bow. However, increasing the hair tension tends to reduce the nonlinearity. On the contrary, different amounts of camber lead to very different profiles of the compliance along the bow. The nonlinearity of the compliance near the tip is especially apparent with increasing camber.

It remains to be seen if results from this study have direct applications in predicting a player's perception of bow performance. Compliance is generally assumed to be of particular importance for bow force control: if too low, the bow might be too sensitive to small variations of hand movement; if too high, the bow would not be responsive enough [START_REF] Askenfelt | Observations on the violin bow and the interaction with the string[END_REF] .

Measurements and playing tests of bows with different settings of camber and hair tension will be performed to go deeper into this question. If significant trends emerge, the model presented in this paper would be useful to assist bow makers. Furthermore, since the bow is often tilted towards the fingerboard in playing [START_REF] Schoonderwaldt | Mechanics and acoustics of violin bowing: Freedom, constraints and 0[END_REF]35 , a model based on spatial beam elements is currently developed in order to take into account the out-of-plane bending of the stick.

FIG. 1 .

 1 FIG. 1. Modern violin bow.

FFIG. 2 .

 2 FIG. 2. Bow without tension (a), tightened (b) and loaded (c).

FIG. 3 .

 3 FIG. 3. Initial and current configurations of a beam element.

  FIG. 5. Bow used for validating the model. (a) Initial measured shape. (b) Successive shapes of the bow from zero tension to high playing tension, by steps of two turns of the button. The y-coordinate indicates the distance from the level of the bow hair to the neutral axis of the stick. (c) Final measured shape. (d) Estimated hair tension T 0 at each level vs. minimum hair-stick distance.

.FIG. 6 .

 6 FIG. 6. Geometry of the standard bow. (a) Tapering, --after Vuillaume 30 , -from Eq. (10). (b) Straight stick. (c) Cambered stick showing the case of a "full camber".

FIG. 7 .

 7 FIG. 7. Variation in simulated hair tension for three values of normal force F y (0.5 N, 1.0 N and 1.5 N) applied at different places along the bow hair, shown as function of (a) relative abscissa γ, and (b) force at the tip γ F y . The simulations are performed on the standard bow with full camber. The initial hair tension T 0 is fixed at 45.0 N, which corresponds to 10 mm hair-stick distance.

FIG. 8 .

 8 FIG. 8. Evolution of deflection at the loading point vs. relative abscissa for applied normal force F y = 1 N. -total deflection, --hair deflection, -•-stick deflection. Standard bow with full camber, initial hair tension T 0 = 45 N.

9 FIG. 9 .

 99 FIG. 9. Evolution of transverse bow deflection at the loading point vs. normal force for three values of relative abscissa. The deviation from a linear evolution (dotted lines) indicates a nonlinear transverse compliance.

FIG. 10 .

 10 FIG. 10. Evolution of minimum hair-stick distance vs. hair tension for four settings of camber (4 mm, 2 mm , 0 mm = full camber and -2 mm). Points denoted by letters (G) to (J) represent playing tension T play 0

  FIG.11. Transverse compliance along the standard bow with full camber for four settings of initial hair tension T 0 , for two normal forces: around 0 N (--) and 1.5 N (-). The four cases correspond to points (0) -(3) in Figure10.

  FIG. 12. Transverse compliance along the bow for different settings of camber (characterized by minimal hair-stick distance without hair tension κ = 4 mm, 2 mm, 0 mm and -2 mm) and playing tension T play 0 , for two normal forces: around 0 N (--) and 1.5 N (-).

  control in performance", Ph.D. thesis, Royal Institute of Technology (KTH), Stockholm, Sweden (2009).

6 FIG. 2 6 FIG. 3 7 FIG. 4 10 FIG. 5 12 FIG. 6 15 FIG. 7 18 FIG. 8 20 FIG. 9 21 FIG. 10 22 FIG. 11

 62637410512615718820921102211 FIG. 1 Modern violin bow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Acknowledgments

The study presented in this paper was partially supported by the French National Research Agency (ANR) within the PAFI project. The authors wish to thank two anonymous reviewers for helpful comments on a previous version of this manuscript, and Matthew Boucher for his careful reading of the manuscript.

imum hair-stick distance κ, and maximal tension T max 0 (tension for straight stick


Christelle Caillouet

Tahiry

Tahiry Razafindralambo

Compromis énergie-délai pour la collecte de données dans les réseaux de capteurs

Keywords: . inria-00586681 optimisation multi-objectif, programmation linéaire, réseaux de capteurs, placement

Les réseaux de capteurs sont utilisés afin de collecter un grand nombre de données de l'environnement. Ces données sont délivrées au travers de l'interface de communication sans fil des capteurs par des communications multi-sauts jusqu'au puits traitant l'information. Le placement de ce puits influe sur les performances des réseaux de capteurs en ce qui concerne le délai et la consommation énergétique, en particulier celle des capteurs intermédiaires servant de relais. L'optimisation de la collecte des données est donc un aspect important dans l'étude des réseaux de capteurs sans fil. Notre article s'interesse à la collecte de données utilisant des puits mobiles. Nous proposons un modèle permettant d'étudier le compromis entre la consommation énergétique du réseau et le délai de collecte des données.

Introduction

Wireless sensor networks (WSNs) have received a lot of attention in recent years due to their potential applications in various areas such as environment monitoring or tracking [START_REF] Barrenetxea | The hitchhiker's guide to successful wireless sensor network deployments[END_REF]. In order to get useful and up-to-date information from the environment, the WSN is composed of a large number of low-capacity (processor, memories, battery) sensors. Unfortunately, the amount of data in the network increases with the number of sensors. The data has to be sent to a central entity, called sink, for storage and processing. Thanks to the wireless communication capabilities and the protocols developed, multi-hop transmissions can be used to route data from a sensor to the sink if no direct connection is available. However, this classical N to 1 communication paradigm rapidly consumes the energy of intermediate sensors and provides unfair delay distribution depending on the distance to the sink [START_REF] Khadar | Performance evaluation of gradient routing strategies for wireless sensor networks[END_REF]. Data collection is therefore a key issue in WSNs.

Various solutions have been proposed to extend the network lifetime and reduce delay for data collection. In [START_REF] Capone | Optimal placement of multiple interconnected gateways in heterogeneous wireless sensor networks[END_REF], authors present an integer linear program for placing a minimum number of gateways and ensure connectivity among them and the sink to form a wireless mesh network to deliver the data. The use of mobile sinks instead of static sinks to collect the data is more efficient and significantly increases the lifetime of the WSN [START_REF] Luo | Joint mobility and routing for lifetime elongation in wireless sensor networks[END_REF]. In their work, the location of the mobile sinks is periodically computed so that the network lifetime is maximized. Some research efforts have focused on approaches either minimizing the energy consumed by the sensors [START_REF] Gandham | Energy efficient schemes for wireless sensor networks with multiple mobile base stations[END_REF], or maximizing the global network lifetime [START_REF] Saad | Multiple mobile sinks positioning in wireless sensor networks for buildings[END_REF]. Considering the route of the mobile sinks in WSNs instead of its periodical relocation has not been addressed in previous work to the best of our knowledge. Our purpose is to determine where to place a set of gateways to collect the data of a region in the WSN field, and compute the route of a mobile sink moving along the gateways to gather data from the sensors. To answer these questions, we propose a Multiobjective Linear Program (MLP) that allows to study the trade-offs between the length of the route of a mobile sink associated with a computed gateway placement, and the overall energy consumption in a WSN. The main contribution is to give a multi-criteria vision of the data collection problem in WSNs. As far as we know there is no multiobjective analysis in this subject. Unlike the works proposed in the literature, the results of this paper are twofold. First, we tackle the problem of optimal placement of data gateways in an energy-efficient WSN. Second, we optimize the data collection tour by the sink to minimize the delay. † Financée par une bourse Post-doctorale de la Fondation allemande Alexander von Humboldt.

Given a WSN represented by a set of sensor nodes S, we define a set of candidate sites C S for the gateway locations. We want each sensor to be associated with its closest gateway : O i is the vector of ordered reachable gateways for sensor i. If j < k, then dist(i, j) ≤ dist(i, k) and O i ( j) is before O i (k) in the vector, where dist(i, j) is the euclidean distance between i and j. Sets J i are index sets of vector O i .

We assume the routing in the WSN (from sensors to gateways) given so that our model is independent of a specific routing strategy. P is the set of paths between the sensors and the candidate sites. O(p) (resp. D(p)) denotes the source node (resp. the destination) of path p ∈ P. From P, we introduce the binary matrix C to indicate the sensor-gateway connectivity : C ic is 1 if it exists a route between sensor i and gateway c.

The energy model considered for the sensors is based on the first order radio model described in [START_REF] Heinzelman | Energy-efficient communication protocol for wireless micro sensor networks[END_REF]. A sensor consumes ε elec = 50 nJ/bit to run the transmitter or receiver circuitry, ε amp = 100 pJ/bit/m 2 for the transmitter amplifier. Thus, to receive a k-bit message, sensor i consumes E r = ε elec k, and E t = ε elec k + ε amp dist 2 (i, j)k to transmit this message to a neighbor j.

E max is a decision variable of our MLP corresponding to the amount of energy spent by a sensor. We then have the binary variables x i j , y j , χ i j indicating respectively if sensor i is assigned to gateway j, if a gateway is installed at candidate site j, and if two gateways are installed at candidate sites i and j and so that link (i, j) is selected for the route of the mobile sink.

To evaluate the overall quality of our solutions, we use the following metrics :

-MinMaxE ( f 1 ) : Balancing the energy spent by the sensors, i.e. WSN lifetime maximization. From the energy model presented above, we seek to minimize the maximum energy spent by each sensor node. -MinRoute ( f 2 ) : Minimizing the route of the mobile sink between the different installed gateways.

The optimization problem of placing the gateways such that we jointly minimize the length of the mobile sink route, and the energy spent by the sensor nodes is the following :

   (i) min f 1 = E max (ii) min f 2 = ∑ i∈C S ∑ j∈C S dist(i, j)χ i j (1) ∑ j∈C S x i j = 1 ∀i ∈ S
(2)

x i j ≤ C i j y j ∀i ∈ S, j ∈ C S (3)
y O i (k) + ∑ h∈J i ,h>k x iO i (h) ≤ 1 ∀i ∈ S (4) ∑ p ∈ P, i ∈ p | i = O(p) (E r + E t ) x O(p)D(p) + ∑ p ∈ P | i = O(p) E t x iD(p) ≤ E max ∀i ∈ S (5) ∑ i∈C S χ i j = y j , ∑ j∈C S χ i j = y i ∀i, j ∈ C S (6)
Obj. (1) minimizes (i) the maximum energy consumed by the sensors, and (ii) the length of the route of a mobile sink along the placed gateways. Constraints (2) and (3) ensure that each sensor must be associated with an installed gateway. Constraints (4) force each sensor to be assigned to its closest gateway. Constraints (5) minimize the maximum amount of energy spent by the sensors, i.e. the sum of the forwarded traffic from other sensors and its own traffic. Equalities (6) force the mobile sink to visit all the chosen gateways to collect data generated by the sensors.

Performance evaluation

We present results obtained with our MLP on networks of size between 50 and 250 sensors in which the sensor's position is randomly chosen in a unitary square area. We use two policies of candidate site locations. First, we divide the area into equal squares and place one candidate site in the center of each square. In this way, the candidate sites form a regular grid. Second, we choose randomly the location of the candidate sites in the area. We solved the proposed MLP with these instances using IBM Cplex solver on an INTEL Core 2 2.4 GHz with 2 Gb of memory. Each result has been averaged on 10 instances. Combining the two metrics of our MLP is not relevant. Indeed, there exists confliction between route length and energy consumption, i.e., pursuing the optimization of the route length of the mobile sink inextricably damages the performance of energy spent by the sensors. Saving energy enforces to deploy more gateways in the network, thus increasing the length of the route for the mobile sinks. The main idea to study the trade-offs between MinMaxE and MinRoute is to find out the possible non-dominated solutions of the multi-objective optimization problem. A solution is non-dominated if it is not possible to improve one of the metrics without worsening at least one another. The set of non-dominated solutions is the Pareto front, that provides a set of solutions that can be chosen depending on the application requirements. In order to generate solutions on the Pareto front, we use the ε-constraint method that transforms the multiobjective problem into a sequence of parameterized single-objective problems such that the optimum of each single-objective problem corresponds to a Pareto-optimal solution.

Effect of candidate sites and network density

Limiting the energy spent by each sensor increases the length of the mobile sink route (see Fig. 1(b) and 1(a)). Moreover, the number of installed gateways strongly depends on the limit of the energy spent. In particular, when we focus on energy (optimizing only MinMaxE, without any constraints on the number of deployed gateways) the optimal solution minimizes the energy spent by each sensor essentially by limiting its forwarding traffic. The gateway placement thus verifies that each sensor is a neighbor of its associated gateway (when possible). When the size of the network increases, then the energy consumption of the sensor nodes also increases (see Fig. 1(a)). The total traffic is more important, so the sensors have more forwarding traffic to relay which increases their load. When the energy is limited to E max = 2 mJ for each sensor, the length of the mobile sink route also increases with the network size. The average route length of the mobile sink equals respectively 0.33, 0.47, 0.6, and 0.65 for a WSN of 100, 150, 200, and 250 nodes. However, the maximum amount of energy spent by the sensor nodes decreases when the number of candidate sites in the network increases as depicted in Fig. 3. On one hand, placing a gateway reduces the relaying traffic and the energy spent. On the other hand, the route length of the mobile sink increases, especially when the energy consumed by each sensor is low. This assertion is confirmed by Fig. 4 that depicts the number of deployed gateways depending on the maximum energy spent by the sensors. The location of the gateways among the WSN is also important regarding the network lifetime and the delay of data collection.

Sensor's load

We define the sensor's load as the number of paths going through a sensor :

Load(i) = ∑ p∈P |i∈p x O(p)D(p) ,
∀i ∈ S. The most loaded sensor is therefore the one that has the maximum number paths going through it : Load(S ) = max i∈S Load(i). Fig. 4 presents the value of Load(S ) in function of the number of deployed gateways. This confirms that deploying more gateways allows to limit the amount of forwarding traffic at each sensor. Another way of limiting the sensor's load is to fairly deploy the gateways among the WSN. When the candidate sites are regularly placed in the area (i.e. the regular policy), then the load is reduced in comparison to a random placement (see Table 1). Load(S ) is the mean value Load(S ) over the Pareto optimum. This value is always greater when the gateway placement is performed among candidate sites chosen randomly, leading to more loaded sensors.

Conclusion

In this paper, we present a framework for efficient data collection in WSNs. We develop a multi-objective linear program with two functions to evaluate the trade-off between the energy spent by the sensors, and the length of the route of mobile sinks collecting data at gateways that we jointly deploy. We show that the sensors' load decreases with the number of gateways until a given threshold when it remains stable. This allows to save energy in the WSN, but, when the number of deployed gateways is important, the mobile sink has a longer route to perform, therefore increasing the delay of data collection until processing. This trade-off must be taken into consideration for an optimal WSN design. Our work provides solutions that allow decision makers to optimally design the data collection plan in WSN with mobile sinks.

R éf érences

FIGURE 1 :

 1 FIGURE 1: Pareto fronts obtained for the random networks tested.

FIGURE 2 :

 2 FIGURE 2: Results for random networks with 25 candidate sites.

  Different size and |C S | = 9.

	MinMaxE (mJ)	0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6	0	0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 250 sensors 200 sensors 150 sensors 100 sensors	MinMaxE (mJ)	0 0.2 0.4 2.2 2 0.6 0.8 1 1.2 1.8 1.6 1.4	0	0.3	0.6	0.9	1.2	1.5 36 gateways 1.8 25 gateways 16 gateways 9 gateways
				MinRoute (a) MinRoute (b) Different number of candidate sites and |S | = 50.

TABLE 1 :

 1 Comparison between regular and random placement for the candidate sites.

				Topology			Grid placement					Random placement
				|S |	|C S |	f 1 * (mJ)	f 2 *	Dpl. Gtw	Load(S )	f 1 * (mJ)	f 2 *		Dpl. Gtw	Load(S )
				50	9	0.45		0.5	3.5	8.93		0.45	0.21		2.96			13.24
				50	16	0.15		0.4	4.25	8		0.35	0.22		4.63			9.83
				100	9	0.45		0.5	3.33	10.2		0.65	0.28		3.71			21.26
				100	16	0.35		0.4	4.81	9.87		0.65	0.21		4			21
				150	9	0.75		0.5	2.85	17		0.95	0.08		3.05			27.88
				150	16	0.45		0.4	4.24	12.76		0.75	0.01		4.41			20.53
				200	9	0.85		0.5	3.9	16.3		0.95	0.22		3.17			20.79
				200	16	0.25		0.4	4.77	13.5		1.05	0.11		3.53			27.4
	Nb of installed gateways	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15	0	0.5	1	1.5	2	2.5	3 250 nodes 3.5 200 nodes 150 nodes 100 nodes 50 nodes		Nb paths on most loaded sensor	0 5 10 15 20 25 30	1	3	5	7	9	11	13 250 nodes 15 200 nodes 150 nodes 100 nodes 50 nodes
						MinMaxE (mJ)								Nb of installed gateways

(a) Deployed gateways vs. E max . (b) Sensors load vs. gateways.