Nicolas Juillet
email: nicolas.juillet@math.unistra.fr

Mario Sigalotti
email: mario.sigalotti@inria.fr

PLIABILITY, OR THE WHITNEY EXTENSION THEOREM FOR CURVES IN CARNOT GROUPS

Keywords: 2010 Mathematics Subject Classification. 22E25, 41A05, 53C17, 54C20, 58C25 Whitney extension theorem, Carnot group, rigid curve, horizontal curve

published or not. The documents may come

Pliability, or the whitney extension theorem for curves in carnot groups

Nicolas Juillet, Mario

Introduction

Extending functions is a basic but fundamental tool in analysis. Fundamental is in particular the extension theorem established by H. Whitney in 1934, which guarantees the existence of an extension of a function defined on a closed set of a finite-dimensional vector space to a function of class C k , provided that the minimal obstruction imposed by Taylor series is satisfied. The Whitney extension theorem plays a significative part in the study of ideals of differentiable functions (see [START_REF] Malgrange | Ideals of differentiable functions[END_REF]) and its variants are still an active research topic of classical analysis (see for instance [START_REF] Fefferman | A sharp form of Whitney's extension theorem[END_REF]).

Analysis on Carnot groups with a homogeneous distance like the Carnot-Carathéodory distance, as presented in Folland and Stein's monograph [START_REF] Folland | Hardy spaces on homogeneous groups[END_REF], is nowadays a classical topic too. Carnot groups provide a generalization of finite-dimensional vector spaces that is both close to the original model and radically different. This is why Carnot groups provide a wonderful field of investigation in many branches of mathematics. Not only the setting is elegant and rich but it is at the natural crossroad between different fields of mathematics, as for instance analysis of PDEs or geometric control theory (see for instance [START_REF]Dynamics, geometry and analysis on sub-Riemannian manifolds, Volumes I-II[END_REF] for a contemporary account). It is therefore natural to recast the Whitney extension theorem in the context of Carnot groups. As far as we know, the first generalization of a Whitney extension theorem to Carnot groups can be found in [START_REF] Franchi | Rectifiability and perimeter in the Heisenberg group[END_REF][START_REF] Franchi | On the structure of finite perimeter sets in step 2 Carnot groups[END_REF], where De Giorgi's result on sets of finite perimeter is adapted first to the Heisenberg group and then to any Carnot group of step 2. This generalization is used in [START_REF] Kirchheim | Rectifiability and parameterization of intrinsic regular surfaces in the Heisenberg group[END_REF], where the authors stress the difference between intrinsic regular hypersurfaces and classical C 1 hypersurfaces in the Heisenberg group. The recent paper [START_REF] Vodop | Whitney-type theorems on the extension of functions on Carnot groups[END_REF] gives a final statement for the Whitney extension theorem for scalar-valued functions on Carnot groups: The most natural generalization that one can imagine holds in its full strength (for more details, see Section 2).

The study of the Whitney extension property for Carnot groups is however not closed. Following a suggestion by Serra Cassano in [START_REF] Cassano | Some topics of geometric measure theory in Carnot groups[END_REF], one might consider maps between Carnot groups instead of solely scalar-valued functions on Carnot groups. The new question presents richer geometrical features and echoes classical topics of metric geometry. We think in particular of the classification of Lipschitz embeddings for metric spaces and of the related question of the extension of Lipschitz maps between metric spaces. We refer to [START_REF] Balogh | Rectifiability and Lipschitz extensions into the Heisenberg group[END_REF][START_REF] Wenger | Lipschitz extensions into jet space Carnot groups[END_REF][START_REF] Rigot | Lipschitz non-extension theorems into jet space Carnot groups[END_REF][START_REF] Balogh | Lipschitz extensions of maps between Heisenberg groups[END_REF] for the corresponding results for the most usual Carnot groups: Abelian groups R m or Heisenberg groups H n (of topological dimension 2n + 1). In view of Pansu-Rademacher theorem on Lipschitz maps (see Theorem 2.1), the most directly related Whitney extension problem is the one for C 1 H -maps, the so-called horizontal maps of class C 1 defined on Carnot groups. This is the framework of our paper.

Simple pieces of argument show that the Whitney extension theorem does not generalize to every ordered pair of Carnot groups. Basic facts in contact geometry suggest that the extension does not hold for (R n+1 , H n ), i.e., for maps from R n+1 to H n . It is actually known that local algebraic constraints of first order make n the maximal dimension for a Legendrian submanifold in a contact manifold of dimension 2n + 1. In fact if the derivative of a differentiable map has range in the kernel of the contact form, the range of the map has dimension at most n. A map from R n+1 to H n is C 1 H if it is C 1 with horizontal derivatives, i.e., if its derivatives take value in the kernel of the canonical contact form. In particular, a C 1 H -map defined on R n+1 is nowhere of maximal rank. Moreover, it is a consequence of the Pansu-Rademacher theorem that a Lipschitz map from R n+1 to H n is derivable at almost every point with only horizontal derivatives. Again n is their maximal rank. In order to contradict the extendability of Lipschitz maps, it is enough to define a function on a subset whose topological constraints force any possible extension to have maximal rank at some point. Let us sketch a concrete example that provides a constraint for the Lipschitz extension problem: It is known that R n can be isometrically embedded in H n with the exponential map (for the Euclidean and Carnot-Carathéodory distances). One can also consider two 'parallel' copies of R n in R n+1 mapped to parallel images in H n : the second is obtained from the first by a vertical translation. Aiming for a contradiction, suppose that there exists an extending Lipschitz map F . It provides on R n × [0, 1] a Lipschitz homotopy between F (R n × {0}) and F (R n × {1}). Using the definition of a Lipschitz map and some topology, the topological dimension of the range is at least n + 1 and its (n + 1)-Hausdorff measure is positive. This is not possible because of the dimensional constraints explained above. See [START_REF] Balogh | Rectifiability and Lipschitz extensions into the Heisenberg group[END_REF] for a more rigorous proof using a different set as a domain for the function to be extended. The proof in [START_REF] Balogh | Rectifiability and Lipschitz extensions into the Heisenberg group[END_REF] is formulated in terms of index theory and purely (n + 1)-unrectifiability of H n . The latter property means that the (n + 1)-Hausdorff measure of the range of a Lipschitz map is zero. Probably this construction and some other ideas from the works on the Lipschitz extension problem [START_REF] Balogh | Rectifiability and Lipschitz extensions into the Heisenberg group[END_REF][START_REF] Wenger | Lipschitz extensions into jet space Carnot groups[END_REF][START_REF] Rigot | Lipschitz non-extension theorems into jet space Carnot groups[END_REF][START_REF] Balogh | Lipschitz extensions of maps between Heisenberg groups[END_REF] can be adapted to the Whitney extension problem. It is not really our concern in the present article to list the similarities between the two problems, but rather to exhibit a class of ordered pairs of Carnot groups for which the validity of the Whitney extension problem depends on the geometry of the groups. Note that a different type of counterexample to the Whitney extension theorem, involving groups which are neither Euclidean spaces nor Heisenberg groups, has been obtained by A. Khozhevnikov in [START_REF] Kozhevnikov | Metric properties of level sets of differentiable maps on Carnot groups[END_REF]. It is described in Example 2.6.

Our work is motivated by F. Serra Cassano's suggestion in his Paris' lecture notes at the Institut Henri Poincaré in 2014 [START_REF] Cassano | Some topics of geometric measure theory in Carnot groups[END_REF]. He proposes -(i) to choose general Carnot groups G as target space, (ii) to look at C 1 H curves only, i.e., C 1 maps from R to G with horizontal derivatives. As we will see, the problem is very different from the Lipschitz extension problem for (R, G) and from the Whitney extension problem for (G, R). Indeed, both such problems can be solved for every G, while the answer to the extendibility question asked by Serra Cassano depends on the choice of G. More precisely, we provide a geometric characterization of those G for which the C 1 H -Whitney extension problem for (R, G) can always be solved. We say in this case that the pair (R, G) has the C 1 H extension property. Examples of target non-Abelian Carnot groups for which C 1 H extendibility is possible have been identified by S. Zimmerman in [START_REF] Zimmerman | The Whitney extension theorem for C 1 , horizontal curves in H n[END_REF], where it is proved that for every n ∈ N the pair (R, H n ) has the C 1 H extension property. The main component of the characterization of Carnot groups G for which (R, G) has the C 1 H extension property is the notion of pliable horizontal vector. A horizontal vector X (identified with a left-invariant vector field) is pliable if for every p ∈ G and every neighborhood Ω of X in the horizontal layer of G, the support of all C 1 H curves with derivative in Ω starting from p in the direction X form a neighborhood of the integral curve of X starting from p (for details, see Definition 3.4 and Proposition 3.7). This notion is close but not equivalent to the property of the integral curves of X not to be rigid in the sense introduced by Bryant and Hsu in [START_REF] Bryant | Rigidity of integral curves of rank 2 distributions[END_REF], as we illustrate by an example (Example 3.5). We say that a Carnot group G is pliable if all its horizontal vectors are pliable. Since any rigid integral curve of a horizontal vector X is not pliable, it is not hard to show that there exist non-pliable Carnot groups of any dimension larger than 3 and of any step larger than 2 (see Example 3.3). On the other hand, we give some criteria ensuring the pliability of a Carnot group, notably the fact that it has step 2 (Theorem 6.5). We also prove the existence of pliable groups of any positive step (Proposition 6.6).

Our main theorem is the following.

Theorem 1.1. The pair (R, G) has the C 1 H extension property if and only if G is pliable. The paper is organized as follows: in Section 2 we recall some basic facts about Carnot groups and we present the C 1 H -Whitney condition in the light of the Pansu-Rademacher Theorem. In Section 3 we introduce the notion of pliability, we discuss its relation with rigidity, and we show that pliability of G is necessary for the C 1 H extension property to hold for (R, G) (Theorem 3.8). The proof of this result goes by assuming that a non-pliable horizontal vector exists and using it to provide an explicit construction of a C 1 H map defined on a closed subset of R which cannot be extended on R. Section 4 is devoted to proving that pliability is also a sufficient condition (Theorem 4.4). In Section 5 we use our result to extend some Lusin-like theorem proved recently by G. Speight for Heisenberg groups [START_REF] Speight | Lusin approximation and horizontal curves in Carnot groups[END_REF] (see also [START_REF] Zimmerman | The Whitney extension theorem for C 1 , horizontal curves in H n[END_REF] for an alternative proof). More precisely, it is proved in [START_REF] Le Donne | Lusin approximation for horizontal curves in step 2 Carnot groups[END_REF] that an absolutely continuous curve in a group of step 2 coincides on a set of arbitrarily small complement with a C 1 H curve. We show that this is the case for pliable Carnot groups (Proposition 5.2). Finally, in Section 6 we give some criteria for testing the pliability of a Carnot group. We first show that the zero horizontal vector is always pliable (Proposition 6.1). Then, by applying some results of control theory providing criteria under which the endpoint mapping is open, we show that G is pliable if its step is equal to 2.

Whitney condition in Carnot groups

A nilpotent Lie group G is said to be a Carnot group if it is stratified, in the sense that its Lie algebra G admits a direct sum decomposition

G 1 ⊕ • • • ⊕ G s , called stratification, such that [G i , G j ] = G i+j for every i, j ∈ N * with i + j ≤ s and [G i , G j ] = {0} if i + j > s. We recall that [G i , G j ] denotes the linear space spanned by {[X, Y ] ∈ G | X ∈ G i , Y ∈ G j }.
The subspace G 1 is called the horizontal layer and it is also denoted by G H . We say that s is the step of G if G s = {0}. The group product of two elements

x 1 , x 2 ∈ G is denoted by x 1 • x 2 . Given X ∈ G we write ad X : G → G for the operator defined by ad X Y = [X, Y ].
The Lie algebra G can be identified with the family of left-invariant vector fields on G. The exponential is the application that maps a vector X of G into the end-point at time 1 of the integral curve of the vector field X starting from the identity of G, denoted by 0 G . That is, if

γ(0) = 0 G γ(t) = X • γ(t)
then γ(1) = exp(X). We also denote by e tX : G → G the flow of the left-invariant vector field X at time t. Notice that e tX (p) = p • exp(tX). Integral curves of left-invariant vector fields are said to be straight curves.

The Lie group G is diffeomorphic to R N with N = s k=1 dim(G k ). A usual way to identify G and R N through a global system of coordinates is to pull-back by exp the group structure from G to G, where it can be expressed by the Baker-Campbell-Hausdorff formula. In this way exp becomes a mapping of G = G onto itself that is simply the identity.

For any λ ∈ R we introduce the dilation ∆ λ :

G → G uniquely characterized by ∆ λ ([X, Y ]) = [∆ λ (X), ∆ λ (Y )] for any X, Y ∈ G, ∆ λ (X) = λX for any X ∈ G 1 .
Using the decomposition

X = X 1 + • • • + X s with X k ∈ G k , it holds ∆ λ (X) = s k=1 λ k X k . For any λ ∈ R we also define on G the dilation δ λ = exp •∆ λ • exp -1 .
Given an absolutely continuous curve γ : [a, b] → G, the velocity γ(t), which exists from almost every t ∈ [a, b], is identified with the element of G whose associated left-invariant vector field, evaluated at γ(t), is equal to γ(t). An absolutely continuous curve γ is said to be horizontal if γ(t) ∈ G H for almost every t. For any interval I of R, we denote by C 1 H (I, G) the space of all curves φ ∈ C 1 (I, G) such that φ(t) ∈ G H for every t ∈ I.

Assume that the horizontal layer G H of the algebra is endowed with a quadratic norm • G H . The Carnot-Carathéodory distance d G (p, q) between two points p, q ∈ G is then defined as the minimal length of a horizontal curve connecting p and q, i.e.,

d G (p, q) = inf b a γ(t) G H dt |γ : [a, b] → G horizontal, γ(a) = p, γ(b) = q .
Note that d G is left-invariant. It is known that d G provides the same topology as the usual one on G. Moreover it is homogeneous, i.e., d G (δ λ p, δ λ q) = |λ|d G (p, q) for any λ ∈ R.

Observe that the Carnot-Carathéodory distance depends on the norm • G H considered on G H . However all Carnot-Carathéodory distances are in fact metrically equivalent. They are even equivalent with any left-invariant homogeneous distance [START_REF] Folland | Hardy spaces on homogeneous groups[END_REF] in a very similar way as all norms on a finite-dimensional vector space are equivalent.

Notice that d G (p, •) can be seen as the value function of the optimal control problem

               γ = m i=1 u i X i (γ), (u 1 , . . . , u m ) ∈ R m , γ(a) = p, b a u 1 (t) 2 + • • • + u m (t) 2 dt → min, where X 1 , . . . , X m is a • G H -orthonormal basis of G H .
Finally, the space C 1 H ([a, b], G) of horizontal curves of class C 1 can be endowed with a natural C 1 metric associated with (d G , • G H ) as follows: the distance between two curves γ 1 and γ

2 in C 1 H ([a, b], G) is max sup t∈[a,b] d G (γ 1 (t), γ 2 (t)), sup t∈[a,b] γ2 (t) -γ1 (t) G H .
In the following, we will write γ2 -γ1

L : G 1 → G 2 with L • δ G 1 λ = δ G 2 λ • L for any λ ∈ R. Moreover L is a homogeneous homomorphism if and only if exp -1 G 2 •L • exp G 1 is a homogeneous Lie algebra morphism. It is in particular a linear map on G 1 identified with G G 1 .
The first layer is mapped on the first layer so that a homogeneous homomorphism from R to G 2 has the form

L(t) = exp G 2 (tX), where X ∈ G G 2
H . Proposition 2.1 (Pansu-Rademacher Theorem). Let f be a locally Lipschitz map from an open subset U of G 1 into G 2 . Then for almost every p ∈ U, there exists a homogeneous homomorphism L p such that

G 1 ∋ q → δ G 2 1/r f (p) -1 • f (p • δ G 1 r (q)) (1)
tends to L p uniformly on every compact set K ⊂ G 1 as r goes to zero.

Note that in

. Let f ∈ C 1 H (G 1 , G 2 )
where G 1 and G 2 are Carnot groups. Let K ⊂ G 1 be compact. Then there exists a function ω from R + to R + with ω(t) = o(t) at 0 + such that for any p, q ∈ K,

d G 2 f (q), f (p) • Df p (p -1 • q) ≤ ω(d G 1 (p, q)),
where Df p is the Pansu derivative.

Proof. This is a direct consequence the "mean value inequality" by Magnani contained in [START_REF] Magnani | Towards differential calculus in stratified groups[END_REF]Theorem 1.2].

The above proposition hints at the suitable formulation of the C 1 -Whitney condition for Carnot groups. This generalization already appeared in the literature in the paper [START_REF] Vodop | Whitney-type theorems on the extension of functions on Carnot groups[END_REF] by Vodop'yanov and Pupyshev.

Definition 2.3 (C 1

H -Whitney condition).

• Let K be a compact subset of G 1 and consider f : K → G 2 and a map L which associates with any p ∈ K a homogeneous group homomorphism L(p). We say that the C 1 H -Whitney condition holds for (f, L) on K if L is continuous and there exists a function

ω from R + to R + with ω(t) = o(t) at 0 + such that for any p, q ∈ K, d G 2 f (q), f (p) • L(p)(p -1 • q) ≤ ω(d G 1 (p, q)). (2)
• Let K 0 be a closed set of G 1 , and f : K 0 → G 2 , and L such that K 0 ∋ p → L(p) is continuous. We say that the C 1 H -Whitney condition holds for (f, L) on K 0 if for any compact set K ⊂ K 0 it holds for the restriction of (f, L) to K.

Of course, according to

Proposition 2.2, if f ∈ C 1 H (G 1 , G 2 ), then the restriction of (f, Df ) to any closed K 0 satisfies the C 1 H -Whitney condition on K 0 .
In this paper we focus on the case G 1 = R. The condition on a compact set K reads r K,η → 0 as η → 0, where

r K,η = sup τ,t∈K, 0<|τ -t|<η d G 2 (f (t), f (τ ) • exp[(t -τ )X(τ )]) |τ -t| , (3) because for every τ ∈ R one has [L(τ )](h) = exp(hX(τ )) for some X(τ ) ∈ G G 2
H and every h ∈ R. With a slight abuse of terminology, we say that the C 1 H -Whitney condition holds for (f, X) on K.

In the classical setting the Whitney condition is equivalent to the existence of a C 1 map f : R n 1 → R n 2 such that f and D f have respectively restrictions f and L on K. This property is usually known as the C 1 -Whitney extension theorem or simply Whitney extension theorem (as for instance in [START_REF] Evans | Measure theory and fine properties of functions[END_REF]), even though the original theorem by Whitney is more general and in particular includes higher order extensions [START_REF] Whitney | Analytic extensions of differentiable functions defined in closed sets[END_REF][START_REF] Whitney | Differentiable functions defined in closed sets[END_REF] and considers the extension f → f as a linear operator. This theorem is of broad use in analysis and is still the subject of dedicated research. See for instance [START_REF] Brudnyi | Generalizations of Whitney's extension theorem[END_REF][START_REF] Fefferman | A sharp form of Whitney's extension theorem[END_REF][START_REF] Fefferman | Sobolev extension by linear operators[END_REF] and the references therein. Definition 2.4. We say that the pair

(G 1 , G 2 ) has the C 1 H extension property if for every (f, L) satisfying the C 1 H -Whitney condition on some closed set K 0 there exists f ∈ C 1 H (G 1 , G 2 ) which extends f on G 1 and such that D fp = L(p) for every p ∈ K 0 .
We now state the C 1 H -extension theorem that Franchi, Serapioni, and Serra Cassano proved in [START_REF] Franchi | On the structure of finite perimeter sets in step 2 Carnot groups[END_REF]Theorem 2.14]. It has been generalised by Vodop'yanov and Pupyshev in [START_REF] Vodop | Whitney-type theorems on the extension of functions on Carnot groups[END_REF][START_REF] Vodop | Whitney-type theorems on the extension of functions on the Carnot group[END_REF] in a form closer to the original Whitney's result including higher order extensions and the linearity of the operator f → f . Theorem 2.5 (Franchi, Serapioni, Serra Cassano). For any Carnot group G 1 and any d ∈ N, the pair (G 1 , R d ) has the C 1 H extension property. The proof proposed by Franchi, Serapioni, and Serra Cassano is established for Carnot groups of step two only, but is identical for general Carnot groups. It is inspired by the proof in [START_REF] Evans | Measure theory and fine properties of functions[END_REF], that corresponds to the special case

G 1 = R n 1 for n 1 ≥ 1.
Let us mention an example from the literature of non-extension with G 1 = R. This remarkable fact was explained to us by A. Kozhevnikov.

Example 2.6. If G 1 and G 2 are the ultrarigid Carnot groups of dimension 17 and 16 respectively, presented in [START_REF] Le Donne | Ultrarigid tangents of sub-Riemannian nilpotent groups[END_REF] and analysed in Lemma A.2.1 of [START_REF] Kozhevnikov | Metric properties of level sets of differentiable maps on Carnot groups[END_REF], one can construct an example (f, L) satisfying the C 1 H -Whitney condition on some compact K without any possible extension ( f , D f ) on G 1 . For this, one exploits the rarity of C 1 H maps of maximal rank in ultrarigid Carnot groups. The definition of ultrarigid from [21, Definition 3.1] is that all quasimorphisms are Carnot similitudes, i.e., a composition of dilations and left-translations. We do not use here directly the definition of ultrarigid groups but just the result stated in Lemma A.2.1 of [START_REF] Kozhevnikov | Metric properties of level sets of differentiable maps on Carnot groups[END_REF] for G 1 and G 2 . Concretely, let us set

K = {(p 1 , . . . , p 17 ) ∈ G 1 | p 2 = • • • = p 16 = 0, p 1 ∈ [-1, 1], p 17 = p 1 }.
Let the map f be constantly equal to 0 on K and L be the constant projection Λ : G 1 ∋ (q 1 , . . . , q 17 ) → (q 1 , . . . , q 16 ) ∈ G 2 . Lemma A.2.1 in [START_REF] Kozhevnikov | Metric properties of level sets of differentiable maps on Carnot groups[END_REF] applied at the point 0 G 1 implies that the only possible extension of f is the projection L(0) = Λ. But this map vanishes only on {p ∈ G 1 | p 1 = • • • = p 16 = 0}, which does not contain K. It remains us to prove that Whitney's condition holds. In fact for two points p = (x, 0 R 15 , x) and q = (y, 0 R 15 , y) in K, we look at the distance from

f (x) = 0 G 2 to f (p) • L(0)(p -1 • q) = L(0)((x, 0 R 15 , x) -1 • (y, 0 R 15 , y)) = (y -x, 0 R 15 )
on the one side and from p to q on the other side. The first one is |y -x|, up to a multiplicative constant, and when |y -x| goes to zero the second one is c|y -x| 1/3 for some constant c > 0. This proves the C 1 H -Whitney condition for (f, L) on K.

In the present paper we provide examples of ordered pairs

(G 1 , G 2 ) with G 1 = R such that the C 1
H extension property does or does not hold, depending on the geometry of G 2 . We do not address the problem of Whitney extensions for orders larger than 1. A preliminary step for considering higher-order extensions would be to provide a suitable Taylor expansion for

C m H -functions from R to G 2 , in the spirit of what recalled for m = 1 in Proposition 2.2.
Let us conclude the section by assuming that the C 1 H extension property holds for some ordered pair (G 1 , G 2 ) of Carnot groups and by showing how to deduce it for other pairs. We describe here below three such possible implications.

(1) Let S 1 be a homogeneous subgroup of G 1 that admits a complementary group K in the sense of [29, Section 4.1.2]: both S 1 and K are homogeneous Lie groups and the intersection is reduced to {0}. Assume moreover that S 1 is a Carnot group and K is normal, so that one can define canonically a projection π : G 1 → S 1 that is a homogeneous homomorphism. Moreover π is Lipschitz continuous (see [START_REF] Cassano | Some topics of geometric measure theory in Carnot groups[END_REF]Proposition 4.13]). For the rest of the section, we say that S 1 is an appropriate Carnot subgroup of G 1 . It can be easily proved that (S 1 , G 2 ) has the C

(G 1 , G 2 × G ′ 2 )
. As a consequence of Theorem 1.1, we can use these three implications to infer pliability statements. Namely, a Carnot group

G is pliable -i) if (G 0 , G) has the C 1
H extension property for some Carnot group G 0 of positive dimension, ii) if G is the appropriate Carnot subgroup of a pliable Carnot group, iii) if G is the product of two pliable Carnot groups.

Rigidity, Necessary condition for the C 1

H extension property Let us first adapt to the case of horizontal curves on Carnot groups the notion of rigid curve introduced by Bryant and Hsu in [START_REF] Bryant | Rigidity of integral curves of rank 2 distributions[END_REF]. We will show in the following that the existence of rigid curves in a Carnot group G can be used to identify obstructions to the validity of the C 1 H extension property for (R, G).

Definition 3.1 (Bryant, Hsu). Let γ ∈ C 1 H ([a, b], G).
We say that γ is rigid if there exists a neighborhood V of γ in the space

C 1 H ([a, b], G) such that if β ∈ V and γ(a) = β(a), γ(b) = β(b) then β is a reparametrization of γ. A vector X ∈ G H is said to be rigid if the curve [0, 1] ∋ t → exp(tX) is rigid.
A celebrated existence result of rigid curves for general sub-Riemannian manifolds has been obtained by Bryant and Hsu in [START_REF] Bryant | Rigidity of integral curves of rank 2 distributions[END_REF] and further improved in [START_REF] Liu | Shortest paths for sub-Riemannian metrics on rank-two distributions[END_REF] and [START_REF] Agrachev | Abnormal sub-Riemannian geodesics: Morse index and rigidity[END_REF]. Examples of Carnot groups with rigid curves have been illustrated in [START_REF] Golé | A note on Carnot geodesics in nilpotent Lie groups[END_REF] and extended in [START_REF] Huang | Extremals in some classes of Carnot groups[END_REF], where it is shown that, for any N ≥ 6 there exists a Carnot group of topological dimension N having rigid curves. Nevertheless, such curves need not be straight. Actually, the construction proposed in [START_REF] Huang | Extremals in some classes of Carnot groups[END_REF] produces curves which are necessarily not straight.

Following [START_REF] Agrachev | Abnormal sub-Riemannian geodesics: Morse index and rigidity[END_REF] (see also [START_REF] Montgomery | A survey of singular curves in sub-Riemannian geometry[END_REF]), and focusing on rigid straight curves in Carnot groups, we can formulate Theorem 3.2 below. In order to state it, let π : T * G → G be the canonical projection and recall that a curve p : I → T * G is said to be an abnormal path if π • p : I → G is a horizontal curve, p(t) = 0 and p(t)X = 0 for every t ∈ I and X ∈ G H , and, moreover, for every Y ∈ G and almost every t ∈ I,

(4) d dt p(t)Y = p(t)[Z(t), Y ],
where Let us illustrate how the existence of rigid straight curves can be deduced from Theorem 3.2 (one could also prove rigidity by direct computations of the same type as those of Example 3.5 below).

Z(t) = d dt π • p(t) ∈ G H . Theorem 3.2. Let X ∈ G H and assume that p : [0, 1] → T * G is an abnormal path with π • p(t) = exp(tX). If t → exp(tX) is rigid, then p(t)[V, W ] = 0 for every V, W ∈ G H and every t ∈ [0, 1]. Moreover, denoting by Q p(t) the quadratic form Q p(t) (V ) = p(t)[V, [X, V ]] defined on {V ∈ G H | V ⊥ X}, we have that Q p(t) ≥ 0 for every t ∈ [0, 1]. Conversely, if p(t)[V, W ] = 0 for every V, W ∈ G H and every t ∈ [0, 1] and Q p(t) > 0 for every t ∈ [0, 1] then t → exp(tX) is rigid.
One immediately checks that p with p(t)X = p(t)Y = p(t)W 1 = 0 and p(t)W 2 = 1 is an abnormal path such that π • p(t) = exp(tX). The rigidity of t → exp(tX) then follows from Theorem 3.2, thanks to the relation

Q p (Y ) = 1.
An extension of the previous construction can be used to exhibit, for every N ≥ 4, a Carnot group of topological dimension N and step N -1 having straight rigid curves. It suffices to consider the N-dimensional Carnot group with Goursat distribution, that is, the group such that dim G 1 = 2, dim G i = 1 for i = 2, . . . , N -1, and there exist two generators X, Y of G 1 whose only nontrivial bracket relations are

[X, Y ] = W 1 and [Y, W i ] = W i+1 for i = 1, . . . , N -3, where G i+1 = Span(W i ) for i = 1, . . . , N -2.
The following definition introduces the notion of pliable horizontal curve, in contrast to a rigid one.

Definition 3.4. We say that a curve γ ∈ C 1 H ([a, b], G) is pliable if for every neighborhood V of γ in C 1 H ([a, b], G) the set {(β(b), β(b)) | β ∈ V, (β, β)(a) = (γ, γ)(a)} is a neighborhood of (γ(b), γ(b)) in G × G H . A vector X ∈ G H is said to be pliable if the curve [0, 1] ∋ t → exp(tX) is pliable.
We say that G is pliable if every vector X ∈ G H is pliable.

By metric equivalence of all Carnot-Carathéodory distances, it follows that the pliability of a horizontal vector does not depend on the norm

• G H considered on G H .
Notice that, by definition of pliability, in every C 1 H neighborhood of a pliable curve γ : [a, b] → G there exists a curve β with β(a) = γ(a), (β, β)(b) = (γ(b), W ), and W = γ(b). This shows that pliable curves are not rigid. It should be noticed, however, that the converse is not true in general, as will be discussed in Example 3.5. In this example we show that there exist horizontal straight curves that are neither rigid nor pliable.

Example 3.5. We consider the 6-dimensional Carnot algebra G of step 3 that is spaned by

X, Y, Z, [X, Z], [Y, Z], [Y, [Y, Z]]
where X, Y, Z is a basis of G 1 and except from permutations all brackets different from the ones above are zero.

According to [7, Chapter 4] there is a group structure on R 6 with coordinates (x, y, z, z 1 , z 2 , z 3 ) isomorphic to the corresponding Carnot group G such that the vectors of G 1 are the leftinvariant vector fields

X = ∂ x , Y = ∂ y , Z = ∂ z + x∂ z 1 + y∂ z 2 + y 2 ∂ z 3 .
Consider the straight curve [0, 1] ∋ t → γ(t) = exp(tZ) ∈ G. First notice that γ is not pliable, since for all horizontal curves in a small enough C 1 neighbourhood of γ the component of the derivative along Z is positive, which implies that the coordinate z 3 is nondecreasing. No endpoint of a horizontal curve starting from 0 G and belonging to a small enough C 1 neighbourhood of γ can have negative z 3 component.

Let us now show that γ is not rigid either. Consider the solution β of

β(t) = Z(β(t)) + u(t)X(β(t)), β(0) = 0 G .
Notice that the y component of β is identically equal to zero. As a consequence, the same is true for the components z 2 and z 3 , while the x, z and z 1 components of β(t) are, respectively, t 0 u(τ )dτ , t, and t 0 τ 0 u(θ)dθdτ . In order to disprove the rigidity, it is then sufficient to take a nontrivial continuous u :

[0, 1] → R such that 1 0 u(τ )dτ = 0 = 1 0 τ 0 u(θ)dθdτ .
Let us list some useful manipulations which transform horizontal curves into horizontal curves. Let γ be a horizontal curve defined on [0, 1] and such that γ(0) = 0 G .

(T1) For every λ > 0, the curve t ∈ [0, λ] → δ λ • γ(λ -1 t) is horizontal and its velocity at time t is γ(λ -1 t). (T2) For every λ < 0, the curve t ∈ [0, |λ|] → δ λ • γ(|λ| -1 t) is horizontal and its velocity at time t is -γ(|λ| -1 t). (T3) The curve γ defined by γ(t) = γ(1) -1 • γ(1 -t) is horizontal. It starts in 0 G and finishes in γ -1 (1). Its velocity at time t is -γ(1 -t). (T4) If one composes the (commuting) transformations (T2) with λ = -1 and (T3), one obtains a curve with derivative γ(1 -t) at time t. (T5) It is possible to define the concatenation of two curves γ 1 : [0, t 1 ] → G and γ 2 : [0, t 2 ] → G both starting from 0 G as follows: the concatenated curve γ : [0, t 1 + t 2 ] → G satisfies γ(0) = 0 G , has the same velocity as γ 1 on [0, t 1 ] and the velocity of

γ 2 (• -t 1 ) on [t 1 , t 1 + t 2 ]. We have γ(t 1 + t 2 ) = γ 1 (t 1 ) • γ(t 2
) as a consequence of the invariance of the the Lie algebra for the left-translation. A consequence of (T1) and (T2) is that X ∈ G H \ {0} is rigid if and only if λX is rigid for every λ ∈ R \ {0}. Similarly, X ∈ G H is pliable if and only if λX is pliable for every λ ∈ R \ {0}.

Proposition 3.7 below gives a characterization of pliable horizontal vectors in terms of a condition which is apriori easier to check than the one appearing in Definition 3.4. Before proving the proposition, let us give a technical lemma. From now on, we write B G (x, r) to denote the ball of center x and radius r in G for the distance d G and, similarly, B G H (x, r) to denote the ball of center x and radius r in G H for the norm • G H . Lemma 3.6. For any x ∈ G and 0 < r < R, there exists ε > 0 such that if y, z ∈ G and

ρ ≥ 0 satisfy d G (y, 0 G ), d G (z, 0 G ), ρ ≤ ε, then B G (x, r) ⊂ y • δ 1-ρ (B G (x, R)) • z.
Proof. Assume, by contradiction, that for every n ∈ N there exist

x n ∈ B G (x, r), y n , z n ∈ B G (0 G , 1/n) and ρ n ∈ [0, 1/n] such that x n ∈ y n • δ 1-ρn (B G (x, R)) • z n . Equivalently, δ (1-ρn) -1 (y -1 n • x n • z -1 n ) ∈ B G (x, R). However, lim sup n→∞ d G (x, δ (1-ρn) -1 (y -1 n • x n • z -1 n )) ≤ r, leading to a contradiction. Proposition 3.7. A vector V ∈ G H is pliable if and only if for every neighborhood V of the curve [0, 1] ∋ t → exp(tV ) in the space C 1 H ([0, 1], G), the set {β(1) | β ∈ V, (β, β)(0) = (0 G , V )} is a neighborhood of exp(V ). Proof. Let F : C 1 H ([0, 1], G) ∋ β → (β, β)(1) ∈ G × G H and denote by π : G × G H → G the canonical projection.
One direction of the equivalence being trivial, let us take ε > 0 and assume that π • F (U ε ) is a neighbourhood of exp(V ) in G, where

U ε = {β ∈ C 1 H ([0, 1], G) | (β, β)(0) = (0 G , V ), β -V ∞,G H < ε}. We should prove that F (U ε ) is a neighborhood of (exp(V ), V ) in G × G H .
Step 1: As an intermediate step, we first prove that there exists η > 0 such that

B G (exp(V ), η)× {V } is contained in F (U ε ).
Let ρ be a real parameter in (0, 1). Using the transformations among horizontal curves described earlier in this section, let us define a map T ρ :

U ε → C 1 H ([0, 1], G) associating with a curve γ ∈ U ε the concatenation (transformation (T5)) of γ 1 : t → δ ρ • γ(ρ -1 t) on [0, ρ]
obtained by transformation (T1) and a curve γ 2 defined as follows. Consider γ 2,1

: [0, 1 -ρ] ∋ t → δ 1-ρ • γ((1 -ρ) -1 t) (again (T1)). The curve γ 2 is defined from γ 2,1 by γ 2 (t) = γ 1 (ρ) • γ 2,1 (1 -ρ) -1 • δ -1 • γ 2,1 ((1 -ρ) -t) (see transformation (T4)). The derivative of T ρ (γ) at time t ∈ [0, ρ) is γ(ρ -1 t). Its derivative at time ρ + t is γ(1 -(1 -ρ) -1 t) for t ∈ (0, 1 -ρ].
Hence T ρ (γ) is continuous and has derivative γ(1) at limit times ρ -and ρ + , i.e., is a well-defined map from U ε into C 1 H ([0, 1], G). Moreover, T ρ (γ) has the same derivative V = γ(0) at times 0 and 1 and its derivative at any time in [0, 1] is in the set of the derivatives of γ. In particular, T ρ (U ε ) ⊂ U ε .

Notice now that, by construction, the endpoint T ρ (γ)(1) of the curve T ρ (γ) is a function of γ(1) and ρ only. It is actually equal to

F ρ (x) = δ ρ (x) • δ ρ-1 (x) -1 ,
where x = γ(1) (see (T1) and (T4)) . Let x 0 = exp(V ) and γ 0 : t → exp(tV ). We have F ρ (x 0 ) = x 0 because T ρ (γ 0 ) = γ 0 , both curves having derivative constantly equal to V . We prove now that for ρ close enough to 1, the differential of F ρ at x 0 is invertible. Let us use the coordinate identification of G with R N . For every y ∈ G, the limits of δ ρ (y) and δ 1-ρ (y) as ρ tends to 1 are y and 0 G respectively, while Dδ ρ (y) and Dδ ρ-1 (y) converge to Id and 0 respectively. One can check (see, e.g., [7, Proposition 2.2.22]) that the inverse function has derivative -Id at 0 G . Finally the left and right translations are global diffeomorphisms. Collecting these informations and applying the chain rule, we get that DF ρ (x 0 ) tends to an invertible operator as ρ goes to 1. Hence for ρ great enough, F ρ (x 0 ) is a local diffeomorphism.

We know by assumption on V that, for any ε > 0, the endpoints of the curves of U ε form a neighborhood of x 0 . We have shown that this is also the case if we replace U ε by T ρ (U ε ), for ρ close to 1. The curves of T ρ (U ε ) are in U ε and have, moreover, derivative V at time 1. He have thus proved that for every ε > 0 there exists η > 0 such that B G (x 0 , η) × {V } is contained in F (U ε ).

Step 2: Let us now prove that

F (U ε ) is a neighborhood of (x 0 , V ) in G × G H .
Let β be a curve in U ε with β(1) = V and consider for every W ∈ B G H (V, ε) and every ρ ∈ (0, 1) the curve α ρ,W defined as follows: α ρ,W = δ 1-ρ •β((1-ρ) -1 t) on [0, 1-ρ] (transformation (T1)) and αρ,W is the linear interpolation between V and W on

[1 -ρ, 1]. Notice that α ρ,W is in U ε .
Let u ∈ G be the endpoint at time ρ of the curve in G starting at 0 G whose derivative is the linear interpolation between V and W on [0, ρ]. Then (α ρ,W , αρ,W )(1) = (δ 1-ρ (β(1)) • u, W ) and u depends only on V , ρ and W , and not on the curve β. Moreover, u tends to 0 G as ρ goes to 1, uniformly with respect to W ∈ B G H (V, ε). Lemma 3.6 implies that for ρ sufficiently close to 1, for every

W ∈ B G H (V, ε), it holds δ 1-ρ (B G (x 0 , η)) • u ⊃ B G (x 0 , η/2). We proved that B G (x 0 , η/2) × B G H (V, ε) ⊂ F (U ε ), concluding the proof of the proposition.
The main result of this section is the following theorem, which constitutes the necessity part of the characterization of C Proof. Suppose, by contradiction, that there exists V ∈ G H which is not pliable. We are going to prove that (R, G) has not the C 1 H extension property. Let γ(t) = exp(tV ) for t ∈ [0, 1]. Since V is not pliable, it follows from Proposition 3.7 that there exist a neighborhood V of γ in the space C 1 H ([0, 1], G) and a sequence (x n ) n≥1 converging to 0 G such that for every n ≥ 1 no curve β in V satisfies (β(0), β(0)) = (0 G , V ) and β(1) = γ(1) • x n . In particular, there exists a neighborhood Ω of V in G H such that for every

β ∈ C 1 H ([0, 1], G) with (β(0), β(0)) = (0 G , V ) and β(t) ∈ Ω, ∀t ∈ [0, 1],
we have (β(1), β(1)) = (γ(1) • x n , V ) for every n ∈ N. Since lim n→∞ x n = 0 G , we can assume without loss of generality that, for every n ≥ 1,

(5) max{d(δ ρ (x n ) • exp(tV ), exp(tV )) | ρ ∈ [0, 1], t ∈ [-1, 1]} ≤ 2 -n .
By homogeneity and left-invariance, we deduce that for every y ∈ G and every ρ > 0, for every β ∈ C 1 ([0, ρ], G) with (β(0), β(0)) = (y, V ) and

β(t) ∈ Ω, ∀t ∈ [0, ρ], we have (β(ρ), β(ρ)) = (y • γ(ρ) • δ ρ (x n ), V ) for every n ∈ N. Define ρ n = 1 n -1 n+1 = 1 n(n+1)
and xn = δ ρn (x n ) for every n ∈ N. It follows from ( 5) that ( 6) max{d(

x n • exp(tV ), exp(tV )) | t ∈ [-1, 1]} ≤ 2 -n , ∀n ≥ 1.
We introduce the sequence defined recursively by y 0 = 0 G and ( 7)

y n+1 = y n • γ(ρ n ) • xn .
Notice that (y n ) n≥1 is a Cauchy sequence and denote by y ∞ its limit as n → ∞. By construction, for every n ∈ N and every

β ∈ C 1 H ([0, ρ n ], G) with (β(0), β(0)) = (y n , V ) and β(t) ∈ Ω for all t ∈ [0, ρ n ], we have (β(ρ n ), β(ρ n )) = (y n+1 , V ). The proof that the (R, G)
has not the C 1 H extension property is then concluded if we show that the C 1 H -Whitney condition holds for (f, X) on K, where

K = ∪ ∞ n=1 1 - 1 n {1},
and f : K → G and X : K → G H are defined by

f (1 -n -1 ) = y n , X(1 -n -1 ) = V, n ∈ N * ∪ {∞}.
For i, j ∈ N * ∪ {∞}, let

D(i, j) = d G (f (1 -i -1 ), f (1 -j -1 ) • exp[(j -1 -i -1 )X(1 -j -1 )]) = d G (y i , y j • exp[(j -1 -i -1 )V ]).
We have to prove that

D(i, j) = o(j -1 -i -1 ) as i, j → ∞, that is, for every ε > 0, there exists i ε ∈ N * such that D(i, j) < ε|j -1 -i -1 | for i, j ∈ N * ∪ {∞} with i, j > i ε .
By triangular inequality we have

D(i, j) ≤ max(i,j)-1 k=min(i,j) d G (y k+1 • exp[((k + 1) -1 -i -1 )V ], y k • exp[(k -1 -i -1 )V ]).
Notice that

d G (y k+1 • exp[((k + 1) -1 -i -1 )V ], y k • exp[(k -1 -i -1 )V ]) = d G (y k+1 • exp[((k + 1) -1 -i -1 )V ], [y k • γ(ρ k )] • exp[((k + 1) -1 -i -1 )V ]) = d G (x k • exp[((k + 1) -1 -i -1 )V ], exp[((k + 1) -1 -i -1 )V ]),
where the last equality follows from [START_REF] Bonfiglioli | Stratified Lie groups and potential theory for their sub-Laplacians[END_REF] and the invariance of d G by left-multiplication. Thanks to [START_REF] Bianchini | Graded approximations and controllability along a trajectory[END_REF], one then concludes that

d G (y k+1 • exp[((k + 1) -1 -i -1 )V ], y k • exp[(k -1 -i -1 )V ]) ≤ 2 -k .
Hence, D(i, j) ≤ max(i,j)-1

k=min(i,j) 2 -k = o(j -1 -i -1
) and this concludes the proof of Theorem 3.8.

Sufficient condition for the C 1 H extension property

We have seen in the previous section that, differently from the classical case, for a general Carnot group G the suitable Whitney condition for (f, X) on K is not sufficient for the existence of an extension (f, ḟ ) of (f, X) on R. More precisely, it follows from Theorem 3.8 that if G has horizontal vectors which are not pliable, then there exist triples (K, f, X) such that the C 1 H -Whitney condition holds for (f, X) on K but there is not a C 1 H -extension of (f, X). In this next section we prove the converse to the result above, showing that the C 1 H extension property holds when all horizontal vectors are pliable, i.e., when G is pliable.

We start by introducing the notion of locally uniformly pliable horizontal vector.

Definition 4.1. A horizontal vector X is called locally uniformly pliable if there exists a neighborhood U of X in G H such that for every ε > 0, there exists η > 0 so that for every

W ∈ U {(γ, γ)(1) | γ ∈ C 1 H ([0, 1], G), (γ, γ)(0) = (0 G , W ), γ -W ∞,G H ≤ ε} ⊃ B G (exp(W ), η) × B G H (W, η).
Remark 4.2. As it happens for pliability, if X is locally uniformly pliable then, for every λ ∈ R \ {0}, λX is locally uniformly pliable.

We are going to see in the following (Remark 6.2) that pliability and local uniform pliability are not equivalent properties. The following proposition, however, establishes the equivalence between pliability and local uniform pliability of all horizontal vectors. Proposition 4.3. If G is pliable, then all horizontal vectors are locally uniformly pliable.

Proof. Assume that G is pliable. For every V ∈ G H and ε > 0 denote by η(V, ε) a positive constant such that

{(γ, γ)(1) | γ ∈ C 1 H ([0, 1], G), (γ, γ)(0) = (0 G , V ), γ -V ∞,G H ≤ ε} ⊃ B G (exp(V ), η(V, ε)) × B G H (V, η(V, ε)).
We are going to show that there exists ν(V, ε) > 0 such that for every

W ∈ B G H (V, ν(V, ε)) {(γ, γ)(1) | γ ∈C 1 H ([0, 1], G), (γ, γ)(0) = (0 G , W ), γ -W ∞,G H ≤ ε} ⊃ B G exp(W ), η V, ε 2 4 × B G H W, η V, ε 2 4 . ( 8
)
The proof of the local uniform pliability of any horizontal vector X is then concluded by simple compactness arguments (taking any compact neighborhood U of X, using the notation of Definition 4.1).

First fix ν(V, ε) > 0 in such a way that exp(W ) ∈ B G (exp(V ), η(V, ε/2)/4)

for every W ∈ B G H (V, ν(V, ε)).
For every W ∈ G H , every ρ ∈ (0, 1), and every curve γ

∈ C 1 H ([0, 1], G) such that (γ, γ)(0) = (0 G , V ), define γ W,ρ ∈ C 1 H ([0, 1], G) as follows: γ W,ρ (0) = 0 G , γW,ρ (t) = (t/ρ)V + ((ρ -t)/ρ)W for t ∈ [0, ρ], γW,ρ (ρ + (1 -ρ)t) = γ(t) for t ∈ [0, 1]. In particular γ W,ρ (1) = γ W,ρ (ρ) • δ 1-ρ (γ(1)), γW,ρ (1) = γ(1), and γW,ρ -W ∞,G H ≤ γ -V ∞,G H + W -V G H . If V -W G H ≤ ε/2, we then have γW,ρ -W ∞,G H ≤ ε ∀γ such that γ -V ∞,G H ≤ ε 2 .
Since γ W,ρ (ρ) depends on V, W , and ρ, but not on γ, we conclude that, for every

W ∈ B G H (V, ε/2), {(β, β)(1) | β ∈ C 1 H ([0, 1], G), (β, β)(0) = (0 G , W ), β -W ∞,G H ≤ ε} ⊃ γ W,ρ (ρ) • δ 1-ρ B G exp(V ), η V, ε 2 × B G H V, η V, ε 2 . Notice that d G (0 G , γ W,ρ (ρ)) ≤ ρ max( V G H , W G H ). Thanks to Lemma 3.6, for ρ suffi- ciently small, γ W,ρ (ρ) • δ 1-ρ B G exp(V ), η V, ε 2 ⊃ B G exp(V ), η V, ε 2 2 .
Now,

B G exp(V ), η V, ε 2 2 ⊃ B G exp(W ), η V, ε 2 4 whenever W ∈ B G H (V, ν(V, ε)).
Similarly,

B G H V, η V, ε 2 ⊃ B G H W, η V, ε 2 4
,

provided that V -W G H ≤ 3η(V, ε/2)/4.
The proof of ( 8) is concluded by taking ν(V, ε) = min(ν(V, ε), ε/2, 3η(V, ε/2)/4).

We are now ready to prove the converse of Theorem 3.8, concluding the proof of Theorem 1.1.

Theorem 4.4. Let G be a pliable Carnot group. Then (R, G) has the C 1 H extension property. Proof. By Proposition 4.3, we can assume that all vectors in G H are locally uniformly pliable. Note moreover that it is enough to prove the extension for maps defined on compact sets K. The generalisation to closed sets K 0 is immediate because the source Carnot group is R. Let (f, X) satisfy the C 1 H -Whitney condition on K where K is compact. We have to define f on the complementary (open) set R \ K, which is the countable and disjoint union of open intervals. For the unbounded components of R \ K, we simply define f as the curve with constant speed X(i) or X(j) where i = min(K) and j = max(K). For the finite components (a, b) we proceed as follows. We consider

y = δ 1/(b-a) (f (a) -1 • f (b)). We let ε be the smallest number such that {(γ, γ)(1) | γ ∈ C 1 H ([0, 1], G), (γ, γ)(0) = (0 G , X(a)), γ -X(a) ∞,G H ≤ ε ′ } contains (y, X(b)) for every ε ′ > ε. We consider an extension f ∈ C 1 H of f on [a, b] such that ḟ (a) = X(a), ḟ (b) = X(b), and ḟ -X(a) ∞,G H ≤ 2ε. By definition of the C 1 H -Whitney condition, there exists a function ω : R + → R + tending to 0 at 0 such that R(a, b) = d G (f (b), f (a)•exp[(b-a)X(a)]) is smaller than ω(b-a)(b-a) and X(b)-X(a) G H ≤ ω(b-a). Since R(a, b) is equal to (b -a)d G (exp(X(a)), y), we can conclude that (9) d G (exp(X(a)), y) ≤ ω(b -a).
Using the corresponding estimates for R(b, a), we deduce that

(10) d G (exp(-X(b)), y -1 ) ≤ ω(b -a).
By construction f extends f and ḟ = X on the interior of K. We prove now that f is

C 1 H and that ḟ = X on the boundary ∂K of K. It is clear that f is C 1 H on R \ ∂K.
In order to conclude the proof we are left to pick x ∈ ∂K, let x n tend to x, and we must show that f (x n ) and ḟ (x n ) tend to f (x) and X(x) respectively. As f X are continuous on K, we can assume without loss of generality that each x n is in R \ K. Assume for now that x n < x for every n. The connected component (a n , b n ) of R \ K containing x n is either constant for n large (in this case x = b n ) or its length goes to zero as n → ∞. In the first case we simply notice that f | [an,bn] is C 1 by construction. In the second case we can assume that a n < x n < b n and b n -a n goes to zero. As f and X are continuous, f (a n ) and X(a n ) converge to f (x) and X(x) respectively. Inequality [START_REF] Bryant | Rigidity of integral curves of rank 2 distributions[END_REF]

guarantees that d G (exp(X(a n )), δ 1/(bn-an) [f (a n ) -1 • f (b n )]) ≤ ω(b n -a n ) → 0 as n → ∞ and the local uniform pliability of X(x) implies that ḟ | [an,bn] -X(a n ) ∞,G H goes to zero as n → ∞. It follows that ḟ (x n ) -X(a n ) G H and d G ( f (x n ), f (a n ))
go to zero, proving that f (x n ) and ḟ (x n ) tend to f (x) and X(x) respectively. The situation where x n > x for infinitely many n can be handled similarly replacing ( 9) by [START_REF] Evans | Measure theory and fine properties of functions[END_REF].

Application to the Lusin approximation of an absolutely continuous curve

In a recent paper, E. Le Donne and G. Speight prove the following result ([22, Theorem 1.2]). Proposition 5.1 (Le Donne-Speight). Let G be a Carnot group of step 2 and consider a horizontal curve γ :

[a, b] → G. For any ε > 0, there exist K ⊂ [a, b] and a C 1 H -curve γ 1 : [a, b] → G such that L([a, b] \ K) < ε and γ = γ 1 on K.
In the case in which G is equal to the n-th Heisenberg group H n , such result had already been proved in [30, Theorem 2.2] (see also [START_REF] Zimmerman | The Whitney extension theorem for C 1 , horizontal curves in H n[END_REF]Corollary 3.8]). In [START_REF] Speight | Lusin approximation and horizontal curves in Carnot groups[END_REF] G. Speight also identifies a horizontal curve on the Engel group such that the statement of Proposition 5.1 is not satisfied ([30, Theorem 3.2]).

The name "Lusin approximation" for the property stated in Proposition 5.1 comes from the use of the classical theorem of Lusin [START_REF] Lusin | Sur les propriétés des fonctions mesurables[END_REF] in the proof. Let us sketch a proof when G is replaced by a vector space R n . The derivative γ of an absolutely continuous curve γ is an integrable function. Lusin's theorem states that γ coincides with a continuous vector-valued function X : K → R n on a set K of measure arbitrarily close to b -a. Thanks to the inner continuity of the Lebesgue measure, one can assume that K is compact. Moreover K can be chosen so that the Whitney condition is satisfied by (γ| K , X) on K. This is a consequence of the mean value inequality

γ(x + h) -γ(x) -h γ(x) ≤ o(h), (11)
where o(h) depends on x ∈ K. By usual arguments of measure theory, inequality [START_REF] Fefferman | A sharp form of Whitney's extension theorem[END_REF] can be made uniform with respect to x if one slightly reduces the measure of K. The (classical) Whitney extension theorem provides a C 1 -curve γ 1 defined on [a, b] with γ 1 = γ and γ1 = X on K.

The proof in [START_REF] Le Donne | Lusin approximation for horizontal curves in step 2 Carnot groups[END_REF] (and also in [START_REF] Speight | Lusin approximation and horizontal curves in Carnot groups[END_REF]) follows the same scheme as the one sketched above. We show here below how the same scheme can be adapted to any pliable Carnot group. The fact that all Carnot groups of step 2 are pliable and that not all pliable Carnot groups are of step 1 or 2 is proved in the next section (Theorem 6.5 and Proposition 6.6), so that our paper actually provides a nontrivial generalization of Proposition 5.1. The novelty of our approach with respect to those in [START_REF] Le Donne | Lusin approximation for horizontal curves in step 2 Carnot groups[END_REF][START_REF] Speight | Lusin approximation and horizontal curves in Carnot groups[END_REF][START_REF] Zimmerman | The Whitney extension theorem for C 1 , horizontal curves in H n[END_REF] is to replace the classical Rademacher differentiablility theorem for Lipschitz or absolutely continuous curves from R to R n by the more adapted Pansu-Rademacher theorem. Proof. We are going to prove that for any ε > 0 there exists a compact set K ⊂ [a, b] with L([a, b] \ K) < ε such the three following conditions are satisfied:

(1) γ(t) exists and it is a horizontal vector at every t ∈ K;

(2) γ| K is uniformly continuous;

(3) For every ε > 0 there exists η > 0 such that, for every

d G (γ(t + h), γ(t) • exp(h γ(t))) = o(h),
as h goes to zero. Let ε be positive. By Lusin's theorem, one can restrict A to a compact set K 1 ⊂ A such that t → γ(t) is uniformly continuous on K 1 and L(A \ K 1 ) < ε/2. Moreover by classical arguments of measure theory, the functions h → |h| -1 d G (γ(t + h), γ(t) • exp(h γ(t 0 ))) can be bounded by a function that is o(1) as h goes to zero, uniformly in t on some compact set K 2 with L(A \ K 2 ) < ε/2 . In other words for every ε > 0 there exists η such that for t ∈ K 2 and h ∈ [t -η, t + η] it holds

d G (γ(t + h), γ(t) • exp(h γ(t))) ≤ ε|h|.
With K = K 1 ∩ K 2 , the three conditions (1), ( 2), (3) listed above hold true.

Case 2: γ general horizontal curve. Let γ be absolutely continuous on [a, b]. It admits a pathlength parametrisation, i.e., there exists a Lipschitz continuous curve ϕ : [0, T ] → G and a function F : [a, b] → [0, T ], absolutely continuous and non-decreasing, such that γ = ϕ • F . Moreover φ has norm 1 at almost every time. As F is absolutely continuous, for every ε > 0 there exists η such that, for any measurable K, the inequality

L([0, T ] \ K) < η implies L([a, b] \ F -1 (K)) < ε.
Let ε be positive and let η be a number corresponding to ε/2 in the previous sentence. Applying to F the scheme of proof sketched after Proposition 5.1 for n = 1, there exists a compact set

K F ⊂ [a, b] with L([a, b] \ K F ) < ε/2 such
that F is differentiable with a continuous derivative on K F and the bound in the mean value inequality is uniform on K F . For the Lipschitz curve ϕ and for every η > 0, Case 1 provides a compact set K ϕ ⊂ [0, T ] with the listed properties with ε/2 in place ε.

Let K be the compact

K F ∩ F -1 (K ϕ ) and note that L([a, b] \ K) < ε. For t ∈ K it holds |F (t + h) -F (t) -hF ′ (t)| = o(h) and d G (ϕ(F (t) + H), ϕ(F (t)) • exp(H φ(F (t)))) = o(H)
, as h and H go to zero, uniformly with respect to t ∈ K. We also know that t → F ′ (t) and t → φ(F (t)) ∈ G H exist and are continuous on K. It is a simple exercise to compose the two Taylor expansions and obtain the wanted conditions for γ = ϕ • F . Note that the derivative of γ on K is F ′ (t) φ(F (t)), which is continuous on K.

Remark 5.3. A set E ⊂ R n is said 1-countably rectifiable if there exists a countable family of Lipschitz curves f k : R → R n such that

H 1 E \ k f k (R) = 0.
The usual Lusin approximation of curves in R n permits one to replace Lipschitz by C 1 in this classical definition of rectifiability. When R n is replaced by a pliable Carnot group the two definitions still make sense and, according to Proposition 5.2, are still equivalent. Rectifiability in metric spaces and Carnot groups is a very active research topic in geometric measure theory (see [START_REF] Le Donne | Lusin approximation for horizontal curves in step 2 Carnot groups[END_REF] for references).

Conditions ensuring pliability

The goal of this section is to identify conditions ensuring that G is pliable. Let us first focus on the pliability of the zero vector. Proposition 6.1. For every Carnot group G, the vector 0 ∈ G is pliable.

Proof. According to Proposition 3.7, we should prove that for every ε > 0 the set

{β(1) ∈ G | β ∈ C 1 H ([0, 1], G), β ∞,G H < ε, (β, β)(0) = (0 G , 0)} is a neighborhood of 0 G in G. Recall that there exist k ∈ N, V 1 , . . . , V k ∈ G H and t 1 , . . . , t k > 0 such that the map φ : (τ 1 , . . . , τ k ) → e τ k V k • • • • • e τ 1 V 1 (0 G )
has rank equal to dim(G) at (τ 1 , . . . , τ k ) = (t 1 , . . . , t k ) and satisfies φ(t 1 , . . . , t k ) = 0 G (see [START_REF] Sussmann | Some properties of vector field systems that are not altered by small perturbations[END_REF]). Notice that for every ν > 0, the function

φ ν : (τ 1 , . . . , τ k ) →e ντ k V k • • • • • e ντ 1 V 1 (0 G ) = e ν 2 τ k ν V k • • • • • e ν 2 τ 1 ν V 1 (0 G ) = δ ν 2 φ τ 1 ν , . . . , τ k ν
has also rank equal to dim(G) at (τ 1 , . . . , τ k ) = (νt 1 , . . . , νt k ) and satisfies φ ν (νt 1 , . . . , νt k ) = 0 G . Hence, up to replacing t j by νt j and V j by ν 2 V j for j = 1, . . . , k and ν small enough, we can assume that t

1 + • • • + t k < 1 and V j G H < ε for j = 1, . . . , k.
Let O be a neighborhood of (t 1 , . . . , t k ) such that for every (τ 1 , . . . , τ k ) ∈ O we have τ 1 , . . . , τ k > 0 and τ

1 + • • • + τ k < 1. Notice that {φ(τ 1 , . . . , τ k ) | (τ 1 , . . . , τ k ) ∈ O} is a neighborhood of 0 G in G.
We complete the proof of the proposition by constructing, for every τ

= (τ 1 , . . . , τ k ) ∈ O a curve β τ ∈ C 1 H ([0, 1], G) such that (12) βτ ∞,G H < ε, (β τ , βτ )(0) = (0 G , 0), β τ (1) = φ(τ ).
For every X ∈ G H , p ∈ G and r > 0 let us exhibit a curve γ ∈ C 1 H ([0, r], G) such that γ(0) = γ(r) = p, γ(0) = 0, γ(r) = X, and γ ∞,G H = X G H . The curve γ can be constructed by imposing γ(r/2) = -X/2 and by extending γ on [0, r/2] and [r/2, r] by convex interpolation. It is also possible to reverse such a curve by transformation (T4) and connect on any segment [0, r] the point-with-velocity (p, X) with the point-with-velocity (p, 0) by a C 1 H curve γ respecting moreover γ ∞,G H = X G H . Finally just concatenating (transformation (T5)) curves of this type it is possible, for every r > 0, to connect (p, X) and (p, Y ) on [0, r] with a curve

γ r,X,Y ∈ C 1 H ([0, r], G) with γr,X,Y ∞,G H = max( X G H , Y G H ).
We then construct β τ as follows: we fix r = (1 -k j=1 τ j )/k, we impose β τ (0) = 0 G and we define βτ to be the concatenation of the following 2k continuous curves in G H : first take γr,0,V 1 , then the constant equal to V 1 for a time τ 1 , then γr,V 1 ,V 2 , then the constant equal to V 2 for a time τ 2 , and so on up to γr,V k-1 ,V k and finally the constant equal to V k for a time τ k . By construction, β τ ∈ C 1 H ([0, 1], G) and satisfies [START_REF] Fefferman | Sobolev extension by linear operators[END_REF]. Remark 6.2. Let us show that, as a consequence of the previous proposition, pliability and local uniform pliability are not equivalent properties (albeit we know from Proposition 4.3 that pliability of all horizontal vectors is equivalent to local uniform pliability of all horizontal vectors).

Recall that local uniform pliability of a horizontal vector X implies pliability of all horizontal vectors in a neighborhood of X (cf. Definition 4.1). Therefore, if 0 is locally uniformly pliable for a Carnot group G then every horizontal vector of G is pliable (Remark 4.2). Hence 0 cannot be locally uniformly pliable if G is not pliable. The remark is concluded by recalling that non-pliable Carnot groups exist (see Examples 3.3 and 3.5).

Let G be a Carnot group and let X 1 , . . . , X m be an orthonormal basis of G H . Let us consider the control system in G × R m given by ( 13)

     γ = m i=1 u i X i (γ), u = v,
where both u = (u 1 , . . . , u m ) and the control v = (v 1 , . . . , v m ) vary in R m .

Let us rewrite x = (γ, u),

F 0 (x) = m i=1 u i X i (γ) 0 , F i (x) = 0 e i for i = 1, . . . , m,
where e 1 , . . . , e m denotes the canonical basis of R m . System (13) can then be rewritten as

(14) ẋ = F 0 (x) + m i=1 v i F i (x).
For every ū ∈ R m , let F ū : L 1 ([0, 1], R m ) → G×R m be the endpoint map at time 1 for system [START_REF] Franchi | Rectifiability and perimeter in the Heisenberg group[END_REF]

with initial condition (0 G , ū). Notice that if x(•) = (γ(•), u(•)) is a solution of (14) with initial condition (0 G , ū) corresponding to a control v ∈ L 1 ([0, 1], R m ), then γ ∈ C 1 H ([0, 1], G) and γ -m i=1 ūi X i ∞,G H ≤ v 1 .
We can then state the following criterium for pliability.

Proposition 6.3. If the map F ū : L 1 ([0, 1], R m ) → G × R m is open at 0, then the horizontal vector m i=1 ūi X i is pliable. As a consequence, if the restriction of F ū to L ∞ ([0, 1], R m ) is open at 0, when the L ∞ topology is considered on L ∞ ([0, 1], R m ), then m
i=1 ūi X i is pliable. We deduce the following property: if a straight curve is not pliable, then it admits an abnormal lift in T * G. Indeed, if a horizontal vector m i=1 ūi X i is not pliable, then the differential of F ū| L ∞ ([0,1],R m ) at 0 must be singular. Hence (see, for instance, [START_REF] Agrachev | Control theory from the geometric viewpoint[END_REF]Section 20.3] or [START_REF] Trélat | Contrôle optimal[END_REF]Proposition 5.3.3]), there exist

p γ : [0, 1] → T * G and p u : [0, 1] → (R m ) * with (p γ , p u ) = 0 such that ṗγ (t) = - ∂ ∂γ H(γ(t), ū, p γ (t), p u (t), 0), ( 15
) ṗu (t) = - ∂ ∂u H(γ(t), ū, p γ (t), p u (t), 0), ( 16
) 0 = ∂ ∂v H(γ(t), ū, p γ (t), p u (t), 0), ( 17
)
for t ∈ [0, 1], where γ(t) = exp(t m i=1 ūi X i ) and

H(γ, u, p γ , p u , v) = p γ m i=1 u i X i (γ) + p u v.
From ( 17) it follows that p u (t) = 0 for all t ∈ [0, 1]. Equation ( 16) then implies that p γ (t)X i (γ(t)) = 0 for every i = 1, . . . , m and every t ∈ [0, 1]. Moreover, p γ must be different from zero. Comparing ( 4) and [START_REF] Franchi | On the structure of finite perimeter sets in step 2 Carnot groups[END_REF], it follows that p γ is an abnormal path. The control literature proposes several criteria for testing the openness at 0 of an endpoint map of the type F ū| L ∞ ([0,1],R m ) . The test presented here below, taken from [START_REF] Bianchini | Graded approximations and controllability along a trajectory[END_REF], generalizes previous criteria obtained in [START_REF] Hermes | Control systems which generate decomposable Lie algebras[END_REF] and [START_REF] Sussmann | A general theorem on local controllability[END_REF]. Theorem 6.4 (Bianchini and Stefani [START_REF] Bianchini | Graded approximations and controllability along a trajectory[END_REF]Corollary 1.2]). Let M be a C ∞ manifold and V 0 , V 1 , . . . , V m be C ∞ vector fields on M. Assume that the family of vector fields J = {ad k V 0 V j | k ≥ 0, j = 1, . . . , m} is Lie bracket generating. Denote by H the iterated brackets of elements in J and recall that the length of an element of H is the sum of the number of times that each of the elements V 0 , . . . , V m appears in its expression. Assume that every element of H in whose expression each of the vector fields V 1 , . . . , V m appears an even number of times is equal, at every q ∈ M, to the linear combination of elements of H of smaller length, evaluated at q. Fix q 0 ∈ M and a neighborhood Ω of 0 in R m . Let U ⊂ L ∞ ([0, 1], Ω) be the set of those controls v such that the solution of q = V 0 (q) + m i=1 v i V i (q), q(0) = q 0 , is defined up to time 1 and denote by Φ(v) the endpoint q(1) of such a solution. Then Φ(U) is a neighborhood of e V 0 (q 0 ).

The following two results show how to apply Theorem 6.4 to guarantee that a Carnot group G is pliable and, hence, that (R, G) has the C 1 H extension property. Theorem 6.5. Let G be a Carnot group of step 2. Then G is pliable and (R, G) has the C 1 H extension property.

Proof. We are going to apply Theorem 6.4 in order to prove that for every horizontal vector Moreover for every i, j = 1, . . . , m,

[[F 0 , F i ], F j ] = 0, [[F 0 , F i ], [F 0 , F j ]](γ, w) = [X i , X j ](γ) 0 ,
and all other Lie bracket in and between elements of J = {ad k F 0 F i | k ≥ 0, i = 1, . . . , m} is zero since G is of step 2.

In particular all Lie brackets between elements of J in which each of the vector fields F 1 , . . . , F m appears an even number of times is zero.

According to Theorem 6.4, we are left to prove that J is Lie bracket generating. This is clearly true, since Span{F i (γ, w), [F 0 , F i ](γ, w), [[F 0 , F i ], [F 0 , F j ]](γ, w) | i, j = 1, . . . , m} is equal to T (γ,w) (G × R m ) for every (γ, w) ∈ G × R m . We conclude the paper by showing how to construct pliable Carnot groups of arbitrarily large step. Proposition 6.6. For every s ≥ 1 there exists a pliable Carnot group of step s. Proof. Fix s ≥ 1 and consider the free nilpotent stratified Lie algebra A of step s generated by s elements Z 1 , . . . , Z s .

For every i = 1, . . . , s, denote by I i the ideal of A generated by Z i and by J i the ideal [I i , I i ]. Then J = ⊕ s i=1 J i is also an ideal of A. Then the factor algebra G = A/J is nilpotent and inherits the stratification of A. Denote by G the Carnot group generated by G. Let X 1 , . . . , X s be the elements of G H obtained by projecting Z 1 , . . . , Z s . By construction, every bracket of X 1 , . . . , X s in G in which at least one of the X i 's appears more than once is zero. Moreover, G has step s, since

[X 1 , [X 2 , [ • • • , X s ], • • • ] is different from zero.
Let us now apply Theorem 6.4 in order to prove that for every X ∈ G H the endpoint map F u : L ∞ ([0, 1], R s ) → G × R s is open at zero, where u ∈ R s is such that X = s i=1 u i X i . Following the same computations as in the proof of Theorem 6.5, ad k+1 F 0 F i (γ, u) = ad k X X i (γ) 0 , k ≥ 0, i = 1, . . . , s.

In particular the family J = {ad k F 0 F i | k ≥ 0, i = 1, . . . , s} is Lie bracket generating. Moreover, every Lie bracket of elements of J = {ad k+1 F 0 F i | k ≥ 0, i = 1, . . . , s} in which at least one of the elements F 1 , . . . , F s appears more than once is zero.

Consider now a Lie bracket W between h ≥ 2 elements of J . Let k 1 , . . . , k s be the number of times in which each of the elements F 1 , . . . , F s appears in W . Let us prove by induction on h that W is the linear combination of brackets between elements of J in which each F i appears k i times, i = 1, . . . , s. Consider the case h = 2. Any bracket of the type [ad k F 0 F i , F j ], k ≥ 0, i, j = 1, . . . , s, is the linear combination of brackets between elements of J in which F i and F j appear once, as it can easily be proved by induction on k, thanks to the Jacobi identity. The induction step on h also follows directly from the Jacobi identity.

We can therefore conclude that every Lie bracket of elements of J in which at least one of the elements F 1 , . . . , F s appears more than once is zero. This implies in particular that the hypotheses of Theorem 6.4 are satisfied, concluding the proof that G is pliable.

Example 3 . 3 .

 33 An example of Carnot structure having rigid straight curves is the standard Engel structure. In this case s = 3, dim G 1 = 2, dim G 2 = dim G 3 = 1 and one can can pick two generators X, Y of the horizontal distribution whose only nontrivial bracket relations are [X, Y ] = W 1 and [Y, W 1 ] = W 2 , where W 1 and W 2 span G 2 and G 3 respectively.

Proposition 5 . 2 (

 52 Lusin approximation of a horizontal curve). Let G be a pliable Carnot group and γ : [a, b] → G be a horizontal curve. Then for any ε > 0 thereexist K ⊂ [a, b] with L([a, b] \ K) < ε and a curve γ 1 : [a, b] → G of class C 1H such that the curves γ and γ 1 coincide on K.

m

  i=1 u i X i the endpoint map F u : L ∞ ([0, 1], R m ) → G × R m is open at zero. Notice that [F 0 , F i ](γ, w) = -X i (γ) 0 , i = 1, . . . , m,and[F 0 , [F 0 , F i ]](γ, w) = m j=1 w j [X i , X j ](γ) 0 , i = 1, . . . , m.

  Proposition 2.1 the map L p is uniquely determined. It is called the Pansu derivative of f at p and denoted by Df p . We denote by C 1 H (G 1 , G 2 ) the space of functions f such that (1) holds at every point p ∈ G 1 and p → Df p is continuous for the usual topology. For G 1 = R this coincides with the definition of C 1 H (I, G 2 ) given earlier. We have the following. Proposition 2.2 (Taylor expansion)

1 H

 1 extension property. In particular, according to[START_REF] Cassano | Some topics of geometric measure theory in Carnot groups[END_REF] Example 4.6], for every k ≤ dim(G G 1 H ) the vector space R k is an appropriate Carnot subgroup of G 1 . Therefore (R k , G 2 ) has the C 1 Assume now that S 2 is an appropriate Carnot subgroup of G 2 . Using the Lipschitz continuity of the projection π : G 2 → S 2 , one easily deduces from the definition of C 1 H -Whitney condition that (G 1 , S 2 ) has the C 1 H extension property. (3) Finally assume that (G 1 , G ′ 2 ) has the C 1 H extension property, where G ′ 2 is a Carnot group. Then one checks without difficulty that the same is true for

	extension property.	H
	(2)

  1 H extendability stated in Theorem 1.1. Theorem 3.8. Let G be a Carnot group. If (R, G) has the C 1 H extension property, then G is pliable.

  t ∈ K and |h| ≤ η with t + h ∈ [a, b], it holds d G (γ(t + h), γ(t) • exp(h γ(t))) ≤ ηε. With these conditions the C 1 H -Whitney condition holds for (γ, γ| K ) on K. Since G is pliable, according to Theorem 4.4 the C 1 H extension property holds for (R, G), yielding γ 1 as in the statement of Proposition 5.2. Case 1: γ is Lipschitz continuous. Let γ be a Lipschitz curve from [a, b] to G. The Pansu-Rademacher theorem (Proposition 2.1) states that there exists A ⊂ [a, b] of full measure such that, for any t ∈ A, the curve γ admits a derivative at t and it holds

Acknowledgment. We warmly thank Frédéric Chapoton and Gwénaël Massuyeau for the suggestions leading us to Proposition 6.6. We are also grateful to Artem Kozhevnikov, Dario Prandi, Luca Rizzi and Andrei Agrachev for many stimulating discussions. This work has been initiated during the IHP trimester "Geometry, analysis and dynamics on sub-Riemannian manifolds" and we wish to thank the Institut Henri Poincaré and the Fondation Sciences Mathématiques de Paris for the welcoming working conditions.

The second author has been supported by the European Research Council, ERC StG 2009 "GeCoMethods", contract number 239748, by the Grant ANR-15-CE40-0018 of the ANR and by the FMJH Program Gaspard Monge in optimization and operation research.


Etienne Malachanne
email: etienne.malachanne@mines-ales.fr

Marie Salgues
email: marie.salgues@mines-ales.fr

Mouad Jebli
email: mouad.jebli@umontpellier.fr

Frederic Jamin
email: frederic.jamin@umontpellier.fr

A COHESIVE ZONE MODEL FOR THE CHARACTERISATION OF THE INTERFACIAL TRANSITION ZONE (ITZ) BETWEEN CEMENT PASTE AND AGGREGATES

Keywords: Interfacial Transition Zone, Tensile strength, Mortar, Cohesive Zone Model

Characterization of concrete behavior needs to know mechanical properties of the two phases constituting them : mortar and aggregates. Nevertheless this bi-phasic approach reaches its limits when concrete leaves the elastic domain. At that stage and according to several studies, phenomena which occur at the interface between mortar and aggregates, or in mortar between cement paste and aggregates, must be taken into account. If the occurence of an Interfacial Transition Zone (ITZ), with weak mechanical properties in regard to the two others surrounding them is well knwon, the modeling of this third phase is not settled yet. This study focus on the characterization of the adhesion at interface between cement paste and aggregates. A mortar compounded by two limestone aggregates binded by a cement paste is considered, and tensile tests are performed on a sample of this composite. Based on these experimental results, a numerical study is developed in order to see influence of interface quality in tensile strength. For thar mortar is modeled by finite elements with a cohesive zone model at the interface, substituting ITZ. With the cohesive zone model used, coupling friction and adhesion at the interface, three parameters have to be fitted : normal and tangential stiffness, and decohesion energy. A strong correlation is found between these parameters and tensile strength, but numerical results show also low values of stiffness and decohesion energy at interface. This result could be explained by a partial adhesion between mortar and cement paste in the sample.

INTRODUCTION

When concrete is considered as a bi-phasic material represented at a mesoscopic scale, compounded by mortar and aggregates, the characterization of its mechanical behavior depends on elastic properties of the two phases, at less until the rupture [START_REF] Prasada | An investigation into the influence of interaggregate spacing and the extent of the itz on properties of portland cement concretes[END_REF][START_REF] Malachanne | Numerical model for mechanical behavior of lightweight concrete and for the prediction of local stress concentration[END_REF]. Nevertheless an acurate prediction when the elastic limit is reached is often investigated. For that, a damage variable may be introduced and the behavior is described by a volumic model as developed by Pijaudier-Cabot and Mazars [START_REF] Pijaudier-Cabot | Damage models for concrete[END_REF]. These models allow to describe the post-peak behavior using inverse analysis methods [START_REF] Comby-Peyrot | Numerical aspects of a problem with damage to simulate mechanical behavior of a quasi-brittle material[END_REF], concrete being still represented as a bi-phasic material. In a different way, some studies consider a third phase, the Interfacial Transition Zone (ITZ) between mortar or cement paste and aggregates, as an independant material with weak mechanical properties [START_REF] Neubauer | A three-phase model of the elastic and shrinkage properties of mortar[END_REF]. In order to take into account this third phase, some numerical studies propose a finite element model representing physically ITZ [START_REF] Lee | A numerical model for elastic modulus of concrete considering interfacial transition zone[END_REF][START_REF] Grondin | How to consider the interfacial transition zones in the finite element modelling of concrete?[END_REF], inducing difficulties particularly in the mesh generation because of the very thin thickness of ITZ in regard to dimensions of other phases [START_REF] Diamond | The itz in concrete a different view based on image analysis and sem observations[END_REF]. In their experimental and numerical studies, Ke et al. [START_REF] Ke | Identification of microstructural characteristics in lightweight aggregate concretes by micromechanical modelling including the interfacial transition zone (itz)[END_REF][START_REF] Ke | Micro-stress analysis and identification of lightweight aggregates failure strength by micromechanical modeling[END_REF] have overcome these difficulties with the development of a micromechanical model, especially for lightweight concrete, in order to predict failure. Other works consider a mechanical damage induced by the quality of the matrix-aggregates interface [START_REF] Keinde | Effect of the interfacial transition zone and the nature of the matrix-aggregate interface on the overall elastic and inelastic behavior of concrete under compression : a 3d numerical study[END_REF]. In the present study we focus on the caracterization of the mechanical link between cement paste and aggregates.

A non perfect adhesion between two phases, may be modeled by introduction of Cohesive Zone Models (CZM) at the interface. These models provide good description of mechanical behavior, also out of elastic domain [START_REF] Elices | The cohesive zone model: advantages, limitations and challenges[END_REF]. Among various studies using CZM in their numerical models, based on finite elements modelling [START_REF] Moroni | Cohesive zone model simulation of fatigue debonding along interfaces[END_REF][START_REF] Elices | Fracture mechanics applied to concrete[END_REF][START_REF] Roesler | Concrete fracture prediction using bilinear softening[END_REF] or with discrete elements [START_REF] De Borst | Mesh-independent discrete numerical representations of cohesive-zone models[END_REF], the one used in this paper, couples adhesion, friction and unilateral contact [START_REF] Raous | A consistent model coupling adhesion, friction, and unilateral contact[END_REF][START_REF] Talon | A model of adhesion coupled to contact and friction[END_REF]. Here, cohesive forces directed at interface are both in tangential and normal directions [START_REF] Bretelle | Unilateral contact with adhesion and friction between two hyperelastic bodies[END_REF]. Cohesion at interface is represented by a damage parameter β, providing the intensity of adhesion.

The work presented here, deals with a characterization of interface quality, between cement paste and limestone aggregate. For that, the cohesive zone model quoted previously is calibrated on mechanical tests performed on a composite compounded by two cubic samples of limestone linked by a thin thickness of cement paste. For that a three-dimensionnal finite elements model of the composite is carried out using the software LMGC90 [START_REF] Dubois | Lmgc90[END_REF]. Parameters of CZM law fitted will describe the mechanical properties and the adhesion quality at the interface between cement paste aggregates.

MATERIALS AND METHODS

Experimental sample of concrete

All experimental results come from works of PhD thesis of Mouad Jebli, from University of Montpellier, France. Composite of mortar is realized with two cubic blocks of limestone binded by a thin cement paste, as represented in the Figure 1. Elastic properties of these two materials are obtained by uniaxial compression tests and are summarized in the Table 1.

Material

Young's modulus (GPa) Poisson ratio Limestone 60 0.3 Cement paste 12.5 0.3 3. This curve will be used on the following to choose parameters of CZM at the interface.

Cohesive Zone Model

Mathematical developments of the model chosen here, using thermodynamic considerations, are detailed on several works [START_REF] Bretelle | Unilateral contact with adhesion and friction between two hyperelastic bodies[END_REF][START_REF] Raous | A consistent model coupling adhesion, friction, and unilateral contact[END_REF]. We will limit in this part to present parameters of the cohesive law, and its main constitutive equations. This model considers a non-penetration between two bodies in contact, here cement paste and aggregates. These two materials are meshed by finite elements. The unilateral contact is coupled with friction, coming from Coulomb law be- tween the two materials, and adhesion governing by a state variable which describes the contact state. This variable β, introduced by Fremond [START_REF] Fremond | Equilibre des structures qui adherent a leur support[END_REF], giving the intensity of adhesion between the two bodies in contact, is defined with a damage parameter D included between 0 (no damage) and 1 (rupture) such as :

β = 1 -D (1)
Thus adhesion linked to damage between aggregate and cement paste is :

• β = 1 total adhesion • 0 < β < 1 partial adhesion • β = 0 no adhesion
The thermodynamic description, based on standard material generalized theory [START_REF] Halphen | Plastic and visco-plastic materials with generalized potential[END_REF], needs to define state variables associated to thermodynamical forces. For that one considers the reversible forces R N and R T associated to the dispalcements u N and u T . Indexes N and T indicate a decomposition into normal and tengential part on the contact surface. Thus, R N and R T are considered as normal and tengential contact forces and the displacement is written :

[u] = [u N ] .n + [u T ] with [u N ] = [u] .n (2)
In a same way, the decomposition of contact forces gives :

R = R N .n + R T with R N = R.n (3)
In this problem the state variables are β, u N and u T , and we introduce a free energy potential defined such as :

ψ(u N , u T , β) = C N 2 u 2 N β 2 + C T 2 u 2 T β 2 -wb (4)
with C N and C T the normal and tengential stiffness, w the Dupre's energy and b the adhesion viscosity coefficient. As explained in works of Raous et al. [START_REF] Raous | A consistent model coupling adhesion, friction, and unilateral contact[END_REF], the main constitutive laws of the contact at the interface come from the subdifferentiation of the free energy potential, and a potential of dissipation, with respect to the state variables u N , u T and β. As consequence we can define the unilateral contact with adhesion :

R N -C N [u N ] β 2 ≥ 0 (5) [u N ] ≥ 0 (R N -C N [u N ] β 2 ) [u N ] = 0
Introducing a fricton coefficient of Coulomb µ, the friction with adhesion is defined by the following equations :

||R T -C T [u T ] β 2 || ≥ µ|R N -C N [u N ] β 2 | (6) ||R T -C T [u T ] β 2 || < µ|R N -C N [u N ] β 2 | ⇒ [ uT ] = 0 ||R T -C T [u T ] β 2 || = µ|R N -C N [u N ] β 2 | ⇒ ∃λ > 0, [ uT ] = λ(R T -C T [u T ] β 2 )
Finally, shape of the cohesive law presented is shown by the curve of the Figure 4 giving the normal reaction with respect to normal displacement. Force-peak represents the limit of adhesion. Indeed at that stage damage occurs at interface, and energy of adhesion, considered as the area under the cruve decreases.

Numerical computation

All the numerical computations have been proceeded with the open software LMGC90 [START_REF] Dubois | Lmgc90[END_REF], developed at the University of Montpellier, France, in which the cohesive zone model presented in past section is implemented. A tree-dimensional finite element model of the composite sample has been carried out. A symmetrical geometry has been chosen, as shown by Figure 5, in order to reduce computation time, thus conditions of symmetry will be respected. Geometry is meshed with hexaedron finite elements H8, and the time step is taken constant and equals to 1 × 10 -4 s. A theta-method is chosen for time intregation with θ = 0.55. Boundary conditions are applied on the lower and upper faces, with a velocity v y = 0.01 mm/s for the tensile load on the upper, and vertical displacements blocked on the lower due to symmetry. Application of the cohesive zone between cement paste and aggregate, is based on contact nodes placed in the contact surface both on cement paste and aggregates. At each time step a detection is realised and rupture occurs when contact is lost between the two bodies. To calibrate cohesive law, several parameters have to be setted : normal and tangential stiffness C N and C T , friction coefficient µ, adhesion energy w and adhesion viscosity b. In this model we consider any viscosity at the interface, thus b = 0. Friction coefficient is determinated by experimental study, and µ = 0.4. We assume that adhesion energy, is defined by the experimental curve stress-strain (Figure 3), and calculated as the aera under the curve, until the rupture. Hence :

w = 1 2 (1.6 × 10 6 × 0.015 × 10 -3 ) = 12 J/m 2 (7)
Hence, adhesion energy calculated here is only the one for the elastic part until rupture, thus according to the Figure 4 presented in the past section we assume that the value corresponding in the cohesive law is near the double, 24 J/m 2 . We will see in the following if this assumption gives results expected.

Finally, only C N and C T have to be fitted with the experimental results. We assume that C N = C T , hence their value will be calibrated with several numerical tensile test on the composite sample.

RESULTS

Numerical computation is performed until the peak-stress, and compared with the experimental stress-strain curve, the objective being to tend to this curve. On the finite element model, we observe that rupture is represented by a decohesion between the two bodies, as shown by the Figure 6.

As explained previously, in this study two parameters have to be fitted : C N , C T . It appears that in our case, C N and C T give the slope of the linear par and the w sets the position of

DISCUSSION

We can first notice a strong correlation between mechanical parameters defined at the interface between cement paste and aggregates, and both strain and tensile strength. Moreover numerical results show that with this cohesive zone model, it is possible to tend to experimental results. We can observe that value of decohesion calculated experimentaly, gives a good estima-tion of the peak stress. The slighly decrease is explained by the interface considered as perfect in the numerical sample which is not the case in the experimental one. As consequence with w fixed by the experiment, only 2 parameters have to be fitted.

Since decohesion occurs only in the ITZ, the latter is substituted by cohesive zones. Thus parameters fitted can be considered as characteristics of the ITZ. A comparison with other studies realised with the same cohesive law, give values of interfacial stiffness higher than the one obtained here. Indeed works of Perales et al. [START_REF] Perales | Two field multibody method for periodic homogenization in fracture mechanics of nonlinear heterogeneous materials[END_REF], have shown that C N = C T = 2 × 10 18 Pa/m when studies of Fouchal et al. [START_REF] Fouchal | Contribution to the modelling of interfaces in masonry construction[END_REF] found 1.5 × 10 17 Pa/m. These values depend on material modeled, hydrided Zircaloy for the first and interface between bricks and mortar for the second. In the same way, our values of decohesion energy are also weak, particularly comparing to adhesion between bricks and mortar (0.9 J/mm 2 ). Two ways may be investigated to explain these low values of interface parameters : parameters identified with numerical computation are not representatives of the real interface properties, or an initial damage occurs at the interface, due to the composite sample fabrication. If the first hypothesis requires other experimental investigations, in particular on a mini-structure of mortar, let us focus on the second through an analysis of Young modulus.

Experimental Young modulus of the composite sample, could be calculated as being the slope of the curve of the Figure 3. Indeed, assuming that until tensile strength, in the linear part, the sample has an elastic behavior, Young modulus is obtained using the Hooke law for an uniaxial tensile test :

E sample = σ ε (8)
with σ the stress and ε the strain. Here, according to the Figure 3, E sample = 3.43 GPa. This value has to be compared with the real Young modulus of the sample, which could be calculated through a serial stiffness. Indeed if we consider that the sample is compounded by three serial elastic bodies, the equivalent stiffness is defined by :

k sample = 1 2 kagg + 1 kp (9)
where k agg and k p are aggregate and paste stiffness defined such as :

k agg,p = S agg,p × E agg,p L agg,p (10)
with S agg = S p = 100 mm 2 the section of aggregate and cement paste, L agg = 15 mm the length of each aggregate and L p = 2 mm the length of the paste. Thus using these values, composite stiffness is k sample = 151515 N/mm leading to the theoretical equivalent Young modulus of the sample :

E th = 32 × 151515 100 = 48.5 GPa (11)
A noticeable difference is observed between theoretical Young modulus and the one obtained experimentally. This result may be explained by a damage at the interface, due to the interface quality between aggregate and cement paste, which lead to a fall on the equivalent Young modulus. According to the definition of the damage variable given by Mazars et al. [START_REF] Pijaudier-Cabot | Damage models for concrete[END_REF], theoretical and experimental Young modulus are binded by the relationship :

E sample = (1 -D)E th ( 12
)
where the damage variable D is different to zero, inducing an intensity of adhesion β less than 1 at the begining of the computation, as explained by equation (2.2). This observation could indicate a partial adhesion between aggregate and cement paste and explain low values of stiffness and decohesion energy at the interface. Thus the strong correlation between stiffness and adhesion energy at interface, and mechanical behavior of the composite, leads to confirm that tensile rupture is in this case governing by the adhesion quality.

CONCLUSION

Cohesive zone model applied to the numerical computation studied here, allows to calibrate decohesion between cement paste and limestone aggregate, using experimental tensile test on a composite sample compounded by both materials. It is possible to reach experimental stress-strain curve, with values of stiffness at the interface and decohesion energy according to literature. Moreover Young modulus obtained by experimental works and numerical computation, calculating the slope of the stress strain curve, is very small against the theoretical one. Other experimental studies have to be performed to confirm the existence of initial damage at the interface.

Figure 1 :

 1 Figure 1: Sample of concrete compounded by two blocks of limestone (10 × 15 × 10 mm) binded by a cement paste (10 × 2 × 10 mm)

Figure 2 :

 2 Figure 2: Tensile test on the concrete sample

Figure 3 :

 3 Figure 3: Stress (in MPa) vs strain for tensile test on the concrete composite

Figure 4 :

 4 Figure 4: Shape of the cohesive law

Figure 5 :

 5 Figure 5: Meshing of a half composite sample by hexaedron finite elements

Figure 6 :

 6 Figure 6: Displacement of the two materials after rupture

Figure 7 :

 7 Figure 7: Influence of parameters C N , C T and w on the mechanical response

Table 1 :

 1 Elastic properties of aggregates and cement paste


Emmanuel Cottanceau

Olivier Thomas
email: olivier.thomas@ensam.eu

Philippe Veron

Marc Alochet

Renaud Deligny

Philippe Véron

A finite

INTRODUCTION

The use of numerical tools for design is crucial in automotive industry. Indeed, they allow to limit the mock-up phases and thus reduce the design duration and the manufacturing costs. Most of the car pieces being rigid body solids, CAD software are nowadays widely used. However, these tools are not able to accurately predict the behavior of flexible pieces. In particular, cables simulation represents an outstanding challenge for the assembly process departments. Their structure is complex : a wire is made up of copper filaments wrapped in an elastomer duct. These wires are most of the time gathered in bundles which are themselves surrounded by various protections such as tape, PVC tube... All these features lead to very complex geometries. In addition, the large displacements undergone by cables, due to their flexibility, bring geometric nonlinearities. Furthermore, design departments require fast computation algorithms such that several design configurations can be tried. The current commercial softwares using computationally costly 3D elements are thus not adapted. In this context, we propose to build a dedicated tool in which the cables are modeled by the geometrically exact beam theory and simulated by an original combination of techniques: the finite element method for discretization, the quaternion parameters to describe rotations and an asymptotic numerical method (ANM) to compute the branches of solution. The slender geometry of cables and their flexibility make a beam model with large displacements/large rotations and small strains really appropriate. These geometrically exact beam models consist in giving the position of the centerline of the beam and the rotation of the sections. They have been widely studied during the last decades but are still of interest for research, mainly because of the difficulty to describe 3D rotations. Indeed, they are not unique and among the main parametrizations existing, most of the papers have been focusing on rotational vector-like formulation, highlighting the use of only 3 parameters [START_REF] Géradin | Flexible Multibody Dynamics[END_REF]. However, it has been proved that at least 4 parameters are necessary to avoid any singularities. Consequently, quaternions based on Euler parameters seem an optimal choice (least parameters avoiding singularities). Beyond their computational efficiency they also offer an alternative description of rotations avoiding trigonometric functions, that we found very well-adapted to be coupled with an ANM.

TECHNIQUES USED Beam formulation

Finite element analysis

Several authors have recently used the quaternions coupled with both finite element methods (FEM) [START_REF] Zupan | On a virtual work consistent 3D Reissner Simo beam formulation using the quaternion algebra[END_REF] and finite difference methods [START_REF]Continuation of equilibria and stability of slender elastic rods using an asymptotic numerical method[END_REF], leading to promising results. For the purposes of our application, we preferred the use of the former. Indeed, the FEM is really convenient for the simulation of complicated geometries, featuring easy assembly of multibody systems (such as piecewise cables). Our FE formulation is obtained from the weak formulation of the beam. As recommended in [START_REF] Zupan | On a virtual work consistent 3D Reissner Simo beam formulation using the quaternion algebra[END_REF], in numerical purposes, we added a Lagrangian multiplier to keep the unity of quaternions, seen as a supplementary variable. In our code, we have used isoparametric elements with a linear interpolation which seems sufficient if enough elements are employed.

Asymptotic Numerical Method

To solve the nonlinear algebraic problem obtained from the FEM f int (u) = λf ext (with internal force vector f int , external force vector f ext , unknowns vector u, load parameter λ), predictor-corrector algorithms are usually used. However, these algorithms require the delicate choice of a step size which has to be small enough to get convergence but not too small to keep efficiency. Besides, the convergence toward the equilibrium solution is never assured. An alternative is to use an asymptotic numerical method [START_REF] Damil | A new method to compute perturbed bifurcation: application to the buckling of imperfect elastic structures[END_REF] [START_REF] Cochelin | A path-following technique via an asymptotic-numerical method[END_REF]. Such a method principle is, starting from a solution point, to seek the branch of solution as an asymptotic expansion of an arc length measure, leading to a semi-analytical solution. The range of validity of the series expansion is automatically computed and allows to define a new starting point for the next branch. The full equilibrium diagram can then be computed iteratively. This method features 3 main advantages : the user does not have any parameter to tune, it requires less computation time than a classical predictor-corrector method and it is not concerned by convergence problems [START_REF] Damil | A new method to compute perturbed bifurcation: application to the buckling of imperfect elastic structures[END_REF]. The main difficulty using this method is to put the system of equations under the quadratic form R(u) = f int (u) -λf ext = C + L(u) + Q(u, u) = 0 (with C, L and Q respectively constant, linear and quadratic operators). However, the use of quaternions instead of the usual trigonometric functions leads to polynomial expressions which makes the quadratization process very straightforward. Using both quaternions and the ANM is thus a very good combination. The full method has been implemented in Matlab. To solve the quadratic system R(u) = 0 with the ANM, use of the free online package ManLab [START_REF] Arquier | Manlab 2.0 : an interactive path-following and bifurcation analysis software[END_REF] has been made. Several standard examples of the literature for which an analytical solution is known [START_REF] Géradin | Flexible Multibody Dynamics[END_REF] [2], as well as examples featuring a more intricate bifurcation diagram have been run. On figure 3, the postbuckling of a clamped-clamped beam is illustrated. On figure 4, the experiment of [START_REF]Continuation of equilibria and stability of slender elastic rods using an asymptotic numerical method[END_REF] is reproduced numerically : it shows the appearance of a plectoneme (loop) on a cable undergoing gravity by the sole rotation of the right end-section. These results demonstrate the accuracy and the robustness of the method for a simple cable (one piece). They will lead our future developments which will consist in testing the method on a great variety of cables and on more complex geometries. In particular, we will focus on bundle-type structures whose cross-sections contain several wires. If concluding, this will allow to simulate accurately the assembly process of cables and will constitute a very useful tool for industry.

RESULTS AND CONCLUSION

Figure 1 :

 1 Figure 1: Cable harnesses.

Figure 2 :

 2 Figure 2: Beam kinematics.

Figure 3 :

 3 Figure 3: Post-buckling of a clamped-clamped beam. Equilibrium curve (left) and corresponding deformed shapes.

Figure 4 :

 4 Figure 4: Formation of a plectoneme [3]. Equilibrium curve (left) and corresponding deformed shapes.