Yacine Chitour

Philippe Jouan

Ronald Manríquez

General Nilpotent and Solvable Approximations of Almost-Riemannian Structures

Keywords: Almost-Riemannian geometry, Nilpotent approximation, Solvable approximation, Genericity

published or not. The documents may come

Introduction

The aim of this paper is twofold. On the rst hand it is shown that the nilpotent or the solvable approximation of an almost-Riemannian structure at a singular point is always a linear almost-Riemannian structure on a Lie group or a homogeneous space. On the other hand the generic almost-Riemannian structures are described and used to exhibit the generic nilpotent and solvable approximations.

More precisely an almost-Riemannian structure, ARS in short, on an ndimensional dierential manifold is a rank-varying sub-Riemannian structure that can be locally dened by a set of n vector elds satisfying the Lie algebra rank condition and such that the singular locus Z, that is the set of points where their rank is not full, is a proper but with empty interior subset (see [START_REF] Agrachev | A Gauss-Bonnet like formula on two-dimensional almost-Riemannian manifolds[END_REF], [START_REF] Agrachev | Two dimensional almost-Riemannian structures with tangency points[END_REF], [START_REF] Ayala | Almost-Riemannian geometry on Lie groups[END_REF], [START_REF] Boscain | Mason Local properties of almost-Riemannian structures in dimension 3[END_REF], [START_REF] Boscain | Normal forms and invariants for 2dimensional almost-Riemannian structures[END_REF], [START_REF] Jouan | Isometries of almost-Riemannian structures on Lie groups[END_REF]).

In particular almost-Riemannian structures on Lie groups or homogeneous spaces that are dened by invariant and linear vector elds only are referred to as linear ARSs (a vector eld on a Lie group is linear if its ow is a one-parameter group of automorphisms)(see [START_REF] Ayala | Almost-Riemannian geometry on Lie groups[END_REF]).

Following [START_REF] Bellaïche | The tangent space in sub-Riemannian geometry in Sub-Riemannian Geometry[END_REF] and [START_REF] Jean | Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion Planning[END_REF] the nilpotent approximation of such a structure at a point p of the singular locus has been used in [START_REF] Boscain | Normal forms and invariants for 2dimensional almost-Riemannian structures[END_REF] and [START_REF] Boscain | Mason Local properties of almost-Riemannian structures in dimension 3[END_REF] to investigate the behaviour around p in dimension respectively 2 and 3.

However it may happen that some vector elds of the nilpotent approximation vanish, changing the almost-Riemannian structure into a sub-Riemannian one. In that case the nilpotent approximation can be replaced by a solvable one, the denition of which was stated in [START_REF] Jouan | Solvable approximations of 3-dimensional almost-Riemannian structures[END_REF] and is recalled in Section 2.2. In [START_REF] Jouan | Solvable approximations of 3-dimensional almost-Riemannian structures[END_REF] the solvable approximation is used to study the distance around some singular points in the generic 3D-case.

The rst aim is herein to prove that the nilpotent or solvable approximation of an ARS at a singular point is a linear almost-Riemannian structure on a Lie group or a homogeneous space, excepted in some very degenerated cases where nor the nilpotent approximation neither the solvable one denes an ARS. It is the purpose of Section 3, and the statements are illustrated by examples postponed to Section 5.

The generic properties of almost-Riemannian structures are then examined in Section 4. It is in particular shown that generically: (1) the singular set Z is a union of submanifolds Z r of codimension r 2 where the rank is n -r; (2) the rank of ∆ + [∆, ∆] is everywhere full (∆ stands for the distribution). The structure of the points of Z r where dim(T p Z r ) + dim(∆ p ) is not maximal is described in Theorem 6. For example in Z 1 these points are the so-called tangency points (see [START_REF] Boscain | Mason Local properties of almost-Riemannian structures in dimension 3[END_REF]), i.e. the points where T p Z 1 = ∆ p .

They are generically isolated in Z 1 .

Thanks to these genericity results and with the help of local normal forms (see Section 4.2) it it nally shown that generically there are only two possibilities for the nilpotent/solvable approximation at a point p ∈ Z:

1. At a tangency point p in Z 1 one vector eld of the nilpotent approximation vanish, but the solvable approximation is not degenerated and denes a linear ARS.

2. At all other points, that is nontangent points of Z 1 and all points in Z r with r ≥ 2, the nilpotent approximation is not degenerated.

In conclusion the only generic points where the solvable approximation is useful are tangency points in Z 1 .

As shown by the examples the picture is very dierent for nongeneric ARSs.

2 Basic denitions 2. [START_REF] Agrachev | A Comprehensive Introduction to Sub-Riemannian Geometry[END_REF] Almost-Riemannian structures For all that concern general sub-Riemannian geometry, including almost-Riemannian one, the reader is referred to [START_REF] Agrachev | A Comprehensive Introduction to Sub-Riemannian Geometry[END_REF].

Let M be a n-dimensional, connected, C ∞ manifold. The C ∞ -module of C ∞ vector elds on M is denoted by Γ(M ). Let ∆ be a submodule of Γ(M ). The ag of submodules

∆ = ∆ 1 ⊆ ∆ 2 ⊆ ∆ 3 ⊆ • • • ⊆ ∆ k ⊆ . . . is dened by induction: ∆ 2 = ∆+[∆, ∆]
is the submodule of Γ(M ) generated by ∆ and the Lie brackets of its elements, and

∆ k+1 = ∆ k + [∆, ∆ k ]. The Lie algebra generated by ∆ is L(∆) = k≥1 ∆ k . The submodule ∆ satises the rank condition if the evaluation of L(∆) at each point q is equal to T q M . Denition 1 An almost-Riemannian structure (resp. distribution) on a smooth n-dimensional manifold M is a triple (E, f, ⟨., .⟩) (resp. a pair (E, f ))
where E is a rank n vector bundle over M , f : E -→ T M is a morphism of vector bundles, and (E, ⟨., .⟩) is an Euclidean bundle, that is ⟨., .⟩ q is an inner product on the ber E q of E, smoothly varying w.r.t. q, assumed to satisfy the following properties: (i) The set of points q ∈ M such that the restriction of f to E q is onto is a proper open and dense subset of M ;

(ii) The module ∆ of vector elds of M , dened as the image by f of the module of smooth sections of E, satises the rank condition.

The set of points of M where the rank of f (E q ) = ∆ q is less than n is called the singular locus of the ARS and denoted by Z.

Denition 2 An almost-Riemannian structure on a smooth n-dimensional manifold M is a pair (∆, ⟨., .⟩) where ∆ is a submodule of Γ(M ) that can be locally dened by n vector elds and satises the rank condition, and ⟨., .⟩ is a bilinear symmetric and positive denite mapping from ∆ × ∆ to C ∞ (M ), such that the set Z of points q where the dimension of ∆ q is less than n is nonempty but with empty interior.

Around any point p ∈ M the submodule ∆ can be locally dened by an orthonormal frame (X 1 , X 2 , . . . , X n ). It is enough to select a set of n sections (e 1 , e 2 , . . . , e n ) of E orthonormal in a neighborhood of p and dene

X i = f * e i where by denition f * e i = f • e i .

Norm

The almost-Riemannian norm on ∆ q is dened by ∥v∥ = min{∥u∥ ; u ∈ E q and f (u) = v}.

Privileged coordinates and approximations

All the material of this section comes from [START_REF] Bellaïche | The tangent space in sub-Riemannian geometry in Sub-Riemannian Geometry[END_REF] and [START_REF] Jean | Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion Planning[END_REF], excepted the solvable approximations that are dened in [START_REF] Jouan | Solvable approximations of 3-dimensional almost-Riemannian structures[END_REF].

Let p be a point of M and let ∆ k (p), k ≥ 1 be the evaluation of the submodule ∆ k at p. Thanks to the rank condition these submodules verify

∆ 1 (p) ⊂ ∆ 2 (p) ⊂ • • • ⊂ ∆ r (p) = T p M
for some integer r referred to as the degree of nonholonomy at p. Let n j stand for the dimension of ∆ j (p). The nonholonomic weights w 1 , w 2 , . . . , w n at p are dened by w i = j ⇐⇒ n j-1 < i ≤ n j .

Let (x 1 , x 2 , . . . , x n ) be a system of coordinates centered at p. These coordinates are privileged if for each i there exist w i vector elds in ∆ such that the Lie derivative X j 1 X j 2 . . . X jw i x i does not vanish at p = 0 but that any such Lie derivative of length smaller than w i vanishes at 0 (see [START_REF] Bellaïche | The tangent space in sub-Riemannian geometry in Sub-Riemannian Geometry[END_REF] or [START_REF] Jean | Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion Planning[END_REF]).

Systems of privileged coordinates always exist (under the rank condition) and in such a system the weighted degree (homogeneous nonholonomic order) of the monomial

x α 1 1 x α 1 2 . . . x αn n is α 1 w 1 + α 2 w 2 + • • • + α n w n ,
the weighted degree of the vector eld ∂ ∂x j is -w j , and the weighted degree of the vector

eld x α 1 1 x α 1 2 . . . x αn n ∂ ∂x j is α 1 w 1 + α 2 w 2 + • • • + α n w n -w j .
More generally the nonholonomic order at p of a function f (resp. a vector eld X) is the minimum of the homogeneous nonholonomic orders of the monomials of its Taylor series.

It is important to notice that the nonholonomic degree ord p X of a vector eld X at p cannot be less than -r, and that ord p [X, Y ] = ord p X + ord p Y if X and Y are homogeneous and [X, Y ] ̸ = 0 (see [START_REF] Bellaïche | The tangent space in sub-Riemannian geometry in Sub-Riemannian Geometry[END_REF] or [START_REF] Jean | Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion Planning[END_REF]).

The nonholonomic order of a vector eld X belonging to ∆ is at least equal to -1. Consider a set X 1 , X 2 , ..., X n of vector elds that generates ∆ around p. In privileged coordinates each X j can be decomposed into

X j = X (-1) j + X (0) j + X (1) j + • • • + X (s) j + . . . where X (s) j
is the component of X j of homogeneous order s.

The nilpotent approximation of

X j ∈ ∆ is X j = X (-1) j .
The Lie algebra generated by X 1 , X 2 , ..., X n is nilpotent and nite dimensional. The rank condition is preserved: it is satised by X 1 , X 2 , ..., X n as soon it is satised by X 1 , X 2 , ..., X n .

It may happen that some of the vector elds X j globally vanish. In that case they can be replaced by X j = X (0) j . Let us assume that only m elements of X 1 , X 2 , ..., X n are linearly independant (as vector elds). As explained in the next section we can assume without lost of generality that these vector elds are X 1 , X 2 , ..., X m and that X m+1 , ..., X n vanish. The set of vector elds X 1 , . . . , X k , , . . . , X m , X m+1 , . . . , X n , is called the solvable approximation of X 1 , X 2 , ..., X n .

ARSs on Lie groups and homogeneous spaces

The purpose of this section is to recall the denition of linear ARSs on Lie groups and homogeneous spaces (see [START_REF] Ayala | Almost-Riemannian geometry on Lie groups[END_REF]).

Let G be a connected Lie group and g its Lie algebra (the set of leftinvariant vector elds, identied with the tangent space at the identity).

A vector eld X on G is linear if its ow is a one-parameter group of automorphisms or equivalently if X (e) = 0 and for any left-invariant eld Y the Lie bracket [X , Y ] is also left-invariant.

A linear ARS on G is an almost-Riemannian structure dened by a set of: n -s left-invariant vector elds Y 1 , . . . , Y n-s . s > 0 linear vector elds X n-s+1 , . . . , X n . assumed to satisfy the rank condition and to have full rank on a proper open and dense subset of G. The almost-Riemannian metric is dened by the orthonormality of this set of vector elds.

The singular set Z where their rank is less than n cannot be empty because at least X n vanishes at the identity. On the other hand its interior is empty by analyticity as soon as the rank is full at one point.

A linear ARS is said to be simple if s = 1. For instance, the famous Grushin plane on the Abelian Lie group R 2 is a simple ARS.

Consider a homogeneous space G/H of G by a closed and connected subgroup H (the elements of G/H are the right cosets of H). The projection of a left-invariant vector eld Y onto G/H is well-dened and will be referred to as a left-invariant vector eld (see details in [START_REF] Ph | Jouan Equivalence of Control Systems with Linear Systems on Lie Groups and Homogeneous Spaces[END_REF]). On the other hand the projection of a linear eld X of G does exist on G/H if and only if H is invariant under its ow, or equivalently, because H is connected, if the Lie algebra of H is ad(X )-invariant. This allows to dene linear vector elds and linear ARSs on homogeneous spaces in exactly the same way than on Lie groups.

3 Nilpotent and solvable approximations are linear Though this section deals with local questions, around a point p belonging to the singular locus, it will be more convenient to assume the ARS dened by a bundle E and a morphism f from E to T M as in Denition 1.

Firstly it is necessary to show that it is always possible to dene the ARS locally, around the point p = 0 in local privileged coordinates, by a set a n orthonormal vector elds X 1 , . . . , X n such that the solvable approximation

X 1 , . . . , X k , X k+1 , . . . , X m , X m+1 , . . . , X n , satises X i (0) ̸ = 0 for i = 1, . . . , k; X i ̸ = 0 but X i (0) = 0 for i = k + 1, . . . , m; X i = 0 for i = m + 1, . . . , n.
Let K p be the kernel of the restriction of f to E p , and let V p be an orthogonal complement to

K p in E p , that is K p ⊥ V p and K p ⊕ V p = E p .
Let e 1 , . . . , e n be a set of n sections of E, orthonormal in a neighborhood of p, such that e j (p) ∈ V p for j = 1, . . . , k and e j (p) ∈ K p for j = k +1, . . . , n.

The vector elds X j = f * (e j ), j = 1, . . . , n dene the ARS around p. Let (x 1 , . . . , x n ) be a set of privileged coordinates and X 1 , . . . , X n be the related nilpotent approximation. Let L be the submodule of Γ(E) generated by e j for j = k + 1, . . . , n. Consider now the mapping e ∈ L -→ f * e.

Its rank is m -k with k ≤ m ≤ n and we can assume without lost of generality that e m+1 , . . . , e n belong to the kernel of that linear map, and that e k+1 , . . . , e m are orthogonal to that kernel.

The vector elds X 1 , . . . , X n satisfy the above conditions.

It may happen that the vector elds X m+1 , . . . , X n fail to be linearly independant. In that case neither the nilpotent approximation nor the solvable one dene an almost-Riemannian structure.

For that reason we will always assume in what follows that the vector elds X 1 , . . . , X k , X k+1 , . . . , X m , X m+1 , . . . , X n , are linearly independant. Denote by Z a (a for approximation) the set of points where their rank is not full. It is not empty because at least one vector eld vanishes at p = 0. On the other hand the approximating vector elds are polynomial and the interior of Z a is empty. Consequently Z a is a proper with empty interior subset of R n and the set of approximating vector elds denes an ARS.

Remarks.

1. It will be shown in the second part of the paper that generically m = n or m = n -1 and that in the second case X n ̸ = 0.

2. In the case where some of the X j vanish or are linearly dependant it seems dicult, if not impossible, to go one step further by considering homogeneous approximations of nonholonomic order s > 0 because, as explained in the sequel, two important properties could be lost. First a homogeneous vector eld of degree s > 0 need not be complete. Second the Lie algebra generated by the approximating vector elds would not be nite dimensional in general. These two drawbacks are related, see [START_REF] Palais | A global formulation of the Lie theory of transformation groups[END_REF] or [START_REF] Ph | Jouan Equivalence of Control Systems with Linear Systems on Lie Groups and Homogeneous Spaces[END_REF].

The generated Lie algebra

In view of the next sections it is very important to notice that all involved vector elds are complete, because of their triangular form. This fact is well-known for the X i (see [START_REF] Bellaïche | The tangent space in sub-Riemannian geometry in Sub-Riemannian Geometry[END_REF] or [START_REF] Jean | Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion Planning[END_REF]).

It is as well true for the order zero homogeneous elds X i . Indeed such a eld writes:

X i = n j=1 P ij (x) ∂ ∂x j
where P ij is polynomial of nonholonomic degree w j . In particular P ij depends linearly on the coordinates (x j 1 , . . . , x js ) of weight w j , is polynomial in the coordinates of weight smaller than w j , and does not depend on the coordinates of weight greater than w j . The associated dierential equation is consequently triangular: the coordinates of weight 1 satisfy a linear homogeneous equation, the coordinates of weight 2 satisfy a linear equation with a second member that depends on the the coordinates of weight 1 only, and so on. All solutions are therefore dened on R and X i is complete. We can state:

Proposition 1 The vector elds X j and X j dened above are complete.

This important property does not hold for homogeneous vector elds of positive degree, for example the rst coordinate such a vector eld could be

x 2 1 ∂ ∂x 1 , but ẋ1 = x 2 1 is not complete.
The second feature we will use in the next subsections is the niteness of the generated Lie algebra; Proposition 2 The Lie algebra L generated by

X 1 , . . . , X k , X k+1 , . . . , X m , X m+1 , . . . , X n , is nite dimensional.
Proof. The nonholonomic order of these vector elds is 0 or -1, and the nonholonomic order of their brackets is in the range -r, . . . , 0 where r is the nonholonomic degree of the set of vector elds. Consequently their components are polynomials of degree less than or equal to r. The Lie algebra L is thus a subspace of a nite dimensional vector space of polynomials. ■

The nilpotent case

It is the case where the vector elds X 1 , . . . , X n are linearly independant and the vectors X 1 (0), . . . , X k (0) are independant in R n . In particular no vector eld X i vanishes, and m = n.

For j = k + 1, . . . , n let D j stand for ad( X j ) and for any multi-index J = (j 1 , ..., j s ) let

D J = D js • • • • • D j 1 (here k + 1 ≤ j u ≤ n and s ≥ 0). Let D = Span{D J ( X i )/ i = 1, . . . , k; J as above}. Lemma 1 The Lie algebra g generated by D is D j -invariant for j = k + 1, . . . , n. Proof. Let D J 1 ( X i 1 ) and D J 2 ( X i 2 ) in D. Then D j [D J 1 ( X i 1 ), D J 2 ( X i 2 )] = [D j •D J 1 ( X i 1 ), D J 2 ( X i 2 )]+[D J 1 ( X i 1 ), D j •D J 2 ( X i 2 )] belongs to g. ■
Let L stand for the Lie algebra generated by X 1 , . . . , X n . It is a wellknown fact that this Lie algebra is nilpotent and nite dimensional (see [START_REF] Bellaïche | The tangent space in sub-Riemannian geometry in Sub-Riemannian Geometry[END_REF] or [START_REF] Jean | Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion Planning[END_REF]).

Theorem 1 1. The ideal generated in L by X 1 , . . . , X k is g. It is a nilpotent Lie algebra.

2. The vector elds X k+1 , . . . , X n do not belong to g and act on g as derivations.

3. The rank at p = 0 of the elements of g is full.

Proof.

1. Since g is D j -invariant for j > k it is clear that it is an ideal of L that contains X 1 , . . . , X k . The ideal generated by these vector elds contains all the D J (X i ) hence is equal to g.

2.

Let us assume that X j belongs to g for some j > k. Because of the rule about the nonholonomic order of brackets of homogeneous vector elds, all the elements of g of order -1 are linear combinations of X 1 , . . . , X k . The vector eld X j is homogeneous of order -1 and can consequently be written as:

X j = k i=1 λ i X i .
But X j vanishes at 0 and the vectors X i (0) are independent by assumption, so that the λ i 's are all equal to 0, a contradiction.

3. Let Y ∈ L \ g. It can be obtained only as brackets of X k+1 , . . . , X n , hence Y (0) = 0 because all these elds vanish at 0.

If g did not satisfy the rank condition then the Lie algebra L neither would satisfy it. This proves item 3.

■ After this analysis at the algebra level we can turn our attention to the Lie group level.

Let G be the simply connected Lie group whose Lie algebra is g. Since g is nilpotent the underlying manifold of G is R N , N = dim(g). The rst task is to show that R n is a homogeneous space of G. This is mainly due to the fact that g is generated by homogeneous vector elds. Lemma 2 The set R n is a homogeneous space of G. More accurately if H stands for the connected subgroup of G whose Lie algebra is the set of elements of g that vanish at 0, then R n is dieomorphic to the quotient

G/H.

Proof.

Since the elements of g are complete vector elds of R n the group G acts naturally on R n as a group of dieomorphisms, and it is enough to show that this action is transitive. More accurately G is the set of exp(t s Y s ) . . . exp(t 1 Y 1 ) where Y i ∈ g is a vector eld on R n and t i ∈ R, i.e. a group of dieomorphisms of R n .

Let us begin by a simple remark. Let Y = n j=1 y j (x) ∂ ∂x j be an element of g, and assume that Y is homogeneous and that Y (0) ̸ = 0. There exists an index i such that y i (0) ̸ = 0, and by homogeneity y i is constant: ∀x ∈ R n , y i (x) = a i ̸ = 0. By homogeneity again the polynomials y j are homogeneous of order w j -w i . Consequently

Y = w j =w i a j ∂ ∂x j + w j >w i y j (x) ∂ ∂x j where a i ̸ = 0.
Since g is generated by homogeneous elements and its rank is full at 0, we can choose in g a set n homogeneous elements Y 1 , Y 2 , . . . , Y n linearly independant at 0. We can also assume that the nonholonomic order at p of Y 1 , . . . , Y n 1 is -1, the one of Y n 1 +1 , . . . , Y n 2 is -2 and so on. Up to linear combinations we can assume that Y 1 , . . . , Y n 1 have the following form:

Y i = ∂ ∂x i + w j >1 y j (x) ∂ ∂x j .
More generally we can assume that if the order of Y i is w i then

Y i = ∂ ∂x i + w j >w i y j (x) ∂ ∂x j .
This way it is clear that the rank of the set of vector elds Y 1 , Y 2 , . . . , Y n is full everywhere in R n . This implies that the action of G on R n is transitive. ■

To complete the construction we associate to the derivation D j = ad( X j ) of g a linear vector eld X j on G for j > k (X j does exist because G is simply connected). It is clear that the projection of X j on R n is X j (see [START_REF] Ph | Jouan Equivalence of Control Systems with Linear Systems on Lie Groups and Homogeneous Spaces[END_REF] for details). Finally the vector elds X 1 , . . . , X k are invariant, and X k+1 , . . . , X n are linear vector elds on the homogeneous space R n = G/H.

We can state:

Theorem 2 The space R n is a homogeneous space of the nilpotent Lie group G whose Lie algebra is g.

The vector elds X 1 , . . . , X k are projections of invariant vector elds of G and X k+1 , . . . , X n are projections of linear vector elds of G.

Consequently the set X 1 , . . . , X n denes a linear ARS on the homogeneous space R n .

3.3

The non-nilpotent case

We set D j = ad( X j ) for j = k + 1, . . . , m and D j = ad( X j ) for j = m + 1, . . . , n. As well as in the nilpotent case we set

D J = D js • • • • • D j 1
for any multi-index J = (j 1 , ..., j s ) where s ≥ 0 and k + 1 ≤ j u ≤ n, and

D = Span{D J (X i )/ i = 1, . . . k; J as above}.
The Lie algebra g generated by D is again D j -invariant for j = k + 1, . . . , n, which shows that g is the ideal generated in L by X 1 , . . . , X k .

Theorem 3 1. The ideal generated in L by X 1 , . . . , X k is g. It is a nilpotent Lie algebra.

2. The vector elds X m+1 , . . . , X n do not belong to g and act on g as derivations.

3. The vector elds X j , with k + 1 ≤ j ≤ m that do not belong to g act on g as derivations.

4. The rank at p = 0 of the elements of g is full.

Proof. The proofs that g is the ideal generated in L by X 1 , . . . , X k and that its rank at p = 0 is full are identical to the nilpotent case.

The Lie algebra g is generated by homogeneous vector elds of order at most -1. Since the order of a Lie bracket is the sum of the orders of the factors and a vector eld of order less than -r vanishes, all brackets of length larger than r vanish, which shows that g is nilpotent.

The points 2. and 3. are clear. ■

Opposite to the nilpotent case we cannot assert that the vector elds X k+1 , . . . , X m do not belong to g. Because of this phenomenon, illustrated by Example 3 in Section 5, we are lead to introduce one more index. Up to a reordering we can assume that X k+1 , . . . , X l belong to g and that X l+1 , . . . , X m do not belong to g, where k + 1 ≤ l ≤ m.

Theorem 4 The space R n is a homogeneous space of the nilpotent Lie group G whose Lie algebra is g.

The vector elds X 1 , . . . , X l are projections of invariant vector elds of G. The vector elds X l+1 , . . . , X m are projections of linear or ane vector elds of G and X m+1 , . . . , X n are projections of linear ones.

Consequently the set of vector elds X 1 , . . . , X m , X m+1 , . . . , X n denes a linear ARS on the homogeneous space R n . Proof. Similar to the one of Theorem 2.

■

Remark.

As shown by Example 4 in Section 5 the Lie algebra L generated by X 1 , . . . , X m , X m+1 , . . . , X n need not be solvable when m ≤ n -2.

However it is solvable if m = n -1 and it will be proven in the next section that generically m = n -1 or m = n.

It is why we call solvable the approximations of the previous kind.

Genericity

The examples of Section 5 show that many dierent, complicated structures may arise and the aim of this section is to determine the generic ones.

In what follows we will say that a property of almost-Riemannian distributions (resp. structures) on a manifold M is generic if for any rank n

vector bundle E (resp. Euclidean vector bundle (E, ⟨., .⟩)) over M the set of smooth morphisms of vector bundles from E to T M for which this property is satised is open and dense in the C 2 Whitney topology.

Let U be an open subset of M on which E and T M are trivializable, and let Π be the projection from E onto M . Then the restriction to Π -1 (U ) of a vector bundle morphism f is equivalent to a smooth mapping X from U to the set M n (R) of n × n square matrices.

Alternately X can be viewed as a mapping (X 1 , X 2 , . . . , X n ) from U to the set Γ(U ) n of n vector elds on U .

It is not useful to assume that f satises the properties of almost-Riemannian distributions because these conditions will turn out to be generic.

The rst two theorems deal with distributions only, and do not require neither metric, nor normal forms.

In what follows we denote by M n×m (R) the set of real n × m matrices (simply M n (R)) if m = n) and by L r the set of elements of M n×m (R) of corank r. It is a submanifold of M n×m (R) of codimension (n-q+r)(m-q+r) where q = min{n, m} (see [START_REF] Golubitsky | Stable mappings and their singularities[END_REF]).

Recall from Section 2 that f being given, ∆ stands for the submodule of Γ(M ) it denes.

4.1

Generic distributions Theorem 5 The following properties are generic:

1. Let R be the largest integer such that R 2 ≤ n. For 1 ≤ r ≤ R let Z r be the set of points where the rank of f p , or locally the rank of {X 1 , X 2 , . . . , X n }, is n -r. Each Z r is a codimension r 2 submanifold and the singular locus Z is the union of these disjoint submanifolds.

2. The submanifold Z r+1 is included in the closure Z r of Z r for r = 1, . . . , R -1.

3. For any local representation X of the distribution the mapping x -→ det(X(x)) is a submersion at all points x ∈ Z 1 .

4. For n ≥ 3 the rank of ∆ + [∆, ∆] is full at all points.

Proof.

1. Let E be a rank n vector bundle over M and let Hom be the vector bundle over M whose ber at p ∈ M is the vector space Hom p = Hom(E p , T p M ) of homomorphisms from E p to T p M .

A smooth vector bundle morphism f from E to T M can be viewed as a smooth section of Hom.

In each ber Hom p the set of morphisms of corank r > 0 is a submanifold of Hom p of codimension r 2 . Their union over M denes a codimension r 2 submanifold S r of Hom. Since the union of the S r for r = 1, . . . , n is closed, the set of smooth sections of Hom that are tranversal to all the S r is open and dense in the set of smooth sections of Hom endowed with the C 2 Whitney topology.

The codimension of S r is r 2 and transversality means nonintersection if r 2 > n. If f -1 (S r ) is not empty transversality implies that it is a submanifold of M of codimension r 2 . This proves 1.

2.

In what follows we will always assume that the distributions under consideration are tranversal to the manifolds S r for r = 1, . . . , n.

Let such a distribution be locally dened by X around a point p assumed to belong to Z r with r ≥ 2. The rank of Λ = X(p) is n-r and there exists an

invertible matrix P such that P ΛP -1 = A Λ B Λ C Λ D Λ where A Λ ∈ M n-r (R)
is invertible. As in the proof of the product of coranks Theorem we consider

T = I 0 -C Λ A -1 Λ I so that T P ΛP -1 = A Λ B Λ 0 D Λ -C Λ A -1 Λ B Λ .
Let the matrix P be xed. The set Ω of matrices Q such that,with obvious notations,

A Q is invertible, hence Φ(Q) = D Q -C Q A -1 Q B Q is well dened, is
open and the mapping Φ is a submersion from this open subset of M n (R) onto M r (R). Clearly rank (Q) = n -r + rank (Φ(Q)) and for 0 ≤ s ≤ r we have L s Mn(R) = Φ -1 (L s Mr(R) ) (with obvious notations again). Since the distribution is generic we have X -⋔ p L r which implies that Φ•X is a submersion at p (see [START_REF] Golubitsky | Stable mappings and their singularities[END_REF], Lemma 4.3). Let V be an open neighborhood of p and

W = V X -1 (Ω). Then Φ • X(W ) is a neighborhood of 0 in M r (R) that encounters L s Mr(R) for 0 ≤ s ≤ r. Consequently ∀s, 0 < s < r, V Z s ̸ = ∅.
3. Consider now the determinant mapping, denoted by det, from

M n (R) to R. The dierential of det at Λ applied to H is d det(Λ).H = Trace( ΛH)
where Λ stands for the transpose of the matrix of cofactors of Λ. Consequently d det(Λ) ̸ = 0 if and only if rank (Λ) ≥ n -1. This shows that det is

a submersion on M n (R) \   r≥2 L r   . In particular det is a submersion in a neighborhood W of L 1 small enough for L 1 = {det = 0} W .
Let us consider a distribution locally dened by X around a point p assumed to belong to Z 1 . Since X -⋔ p L 1 and according to [START_REF] Golubitsky | Stable mappings and their singularities[END_REF] (Lemma 4.3) det •X is a submersion at p. In other words q -→ det(X(q)) is a submersion at p. 4. It remains to show that the rank of ∆ + [∆, ∆] is generically full everywhere. Let X = (X 1 , . . . , X n ) be a set of n vector elds on an open subset U of M that dene locally the distribution. Up to a system of coordinates X is a mapping from U to (R n ) n ≡ M n (R). Let J 1 be a typical ber of the space of 1-jets of sections of T n U , identied with (R n ) n ×(M n (R)) n . The set F i of elements X = (X, dX) of J 1 such that X i = 0 and rank (dX i ) < n is closed with empty interior: it is a nite union of submanifolds of codimension n+ρ 2 , where ρ is the corank of dX i . Consequently the set O = J 1 \( n i=1 F i ) is the open and dense subset of J 1 of elements that verify X i = 0 =⇒ dX i invertible.

Let Ψ be the mapping from

J 1 to M n× n(n+1) 2 (R) dened by Ψ(X) = X, ([X i , X j ] = dX j .X i -dX i .X j ) 1≤i<j≤n .
The mapping Ψ is a submersion on O.

On the other hand the set L r of elements of M n× n(n+1)

2

(R) of corank r is a submanifold of codimension r n(n-1)

2

+ r and Ψ -1 (L r ) O is a submanifold of O of the same codimension.

But n < r n(n-1)

2

+ r excepted in the particular case n = 2 and r = 1 where there is equality (this case has been studied in [START_REF] Agrachev | Two dimensional almost-Riemannian structures with tangency points[END_REF]).

To nish the set of X that are transversal to the F i , the union of which is closed, is open and dense. Such distributions take their values in O, and the set of X that are moreover transversal to all the Ψ -1 (L r ) O is open and dense.

Excepted in the case n = 2 and r = 1 transversality means nonintersection and implies that the rank of ∆ + [∆, ∆] is full at all points. ■ Two subspaces of T p M are attached to a point p belonging to the strate Z r of the singular locus, namely ∆ p , the distribution at p, and T p Z r , the tangent subspace to Z r at p. Their dimensions being respectively n -r and n -r 2 , the dimension of T p Z r + ∆ p is at most equal to min(n, 2n -r 2 -r). We are interested in the cases where the actual dimension of T p Z r + ∆ p is less than min(n, 2n -r 2 -r). For example in Z 1 this means that ∆ p = T p Z r (tangency points).

In what follows we note s = min(n, 2n -r 2 -r) -dim (T p Z r + ∆ p ), and ⌊α⌋ stands for the integer part of the real number α. Theorem 6 The following properties are generic:

r = 1. The points p ∈ Z 1 where T p Z 1 = ∆ p are isolated in Z 1 .
r ≥ 2. Let m(n, r) be the largest dimension that T p Z r + ∆ p may reach, that is m(n, r) = min(n, 2n -r 2 -r), and let s = m(n, r) -dim(T p Z r + ∆ p ). Then 1. The set of points p ∈ Z r where s = 1 is a submanifold of Z r for

n ≥ r 2 + r -⌊ r -1 2
⌋. It is empty if n < r 2 + r -⌊ r -1 2 ⌋.
2. The set of points p ∈ Z r where s ≥ 2 and

s 2 ≤ r is a submanifold of Z r for r 2 + r -⌊ r -s 2 s -1 ⌋ ≤ n ≤ r 2 + r + ⌊ r -s 2 s -1 ⌋. It is empty if n is not in this interval.
3. The set of points p ∈ Z r where s ≥ 2 and s 2 > r is empty.

Proof. In this proof all distributions are assumed to satisfy the transversality conditions of Theorem 5.

1. Let us consider such a distribution and a point p in Z r for some r > 0, locally dened by X on an open set U containing p.

Moreover the set L r is a codimension r 2 submanifold of M n (R) and it can be locally dened, in a neighborhood V of the point X(p) ∈ L r , by a submersion Φ from V to R r 2 , so that L r V = Φ -1 (0).

Let us denote by d(p) the dimension of the quotient (T

p Z r + ∆ p ) /T p Z r .
Since the dimension of Z r is n -r 2 and the one of ∆ p is n -r we have n -r ≤ dim(T p Z r + ∆ p ) ≤ min(n, 2n -r 2 -r), and:

r 2 -r ≤ d(p) ≤ min(n -r, r 2 ).
The distribution being assumed to be transversal to L r at p the restriction of dX at any supplementary subspace to T p Z r in T p U is an isomorphism onto its image. On the other hand dX(p)(T p Z r ) ⊂ ker(dΦ(X p )) because Z r = X -1 (L r ), and since Φ is a submersion at p we obtain

d(p) = rank d(Φ • X)(p).X(p).
Moreover there are exactly n -r indices such that the vector elds X i 1 , X i 2 , . . . , X i n-r are independant at p so that

d(p) = rank (d(Φ • X)(p).X i 1 (p), . . . , d(Φ • X)(p).X i n-r (p)).
3. In a neighborhood of p where the vector elds X i 1 , X i 2 , . . . , X i n-r are independant at all points we dene:

x -→ Θ(x) = (X(x); d(Φ • X)(x).X i 1 (x), . . . , d(Φ • X)(x).X i n-r (x))
The mapping Θ takes its values in

M n (R) × M r 2 ×(n-r) (R) and Θ(x) belongs to L r × L s if and only if x ∈ Z r and d(x) = min(r 2 , n -r) -s.
This shows that s measures the gap between d(p) and the maximal value it can take. Consequently we are interested in the values of s that verify:

0 ≤ s ≤ min(r, n -r 2 ).
4. The sets L r × L s are submanifolds in the typical bers of the space of 1-jets of smooth sections of Hom. Since their union is closed the set of smooth sections of Hom that are transversal to all these submanifolds is open and dense in the set of smooth sections of Hom endowed with the Whitney C 2 topology.

Let us rst assume that

n -r ≥ r 2 . The codimension of L r × L s in M n (R) × M r 2 ×(n-r) (R) is r 2 + s(n -r 2 -r + s).
For s = 1 it is n -r + 1. Since n ≥ n -r + 1 the set of points of Z r where s = 1 is generically a submanifold of codimension n -r + 1. In particular for r = 1 the codimension of this submanifold is n and it consists in isolated points.

For s ≥ 2 transversality means nonintersection if n < r 2 + s(n -r 2r + s). But the two conditions n ≥ r 2 + s(n -r 2 -r + s) and n -r ≥ r 2 are equivalent to 0 ≤ n -r 2 -r ≤ r -s 2 s -1 , hence to:

r 2 + r ≤ n ≤ r 2 + r + ⌊ r -s 2 s -1 ⌋. (1)
Notice that s 2 ≤ r is a necessary condition for this inequality to hold.

Let us now assume that

n -r ≤ r 2 . The codimension of L r × L s is r 2 + s(r 2 + r + s -n). Again transversality means nonintersection if n < r 2 + s(r 2 + r + s -n).
The two conditions n -r ≤ r 2 and n ≥ r 2 + s(r 2 + r + s -n) are equivalent to:

r 2 + r -⌊ r -s 2 s -1 ⌋ ≤ n ≤ r 2 + r (2)
Again s 2 ≤ r is necessary for this inequality to hold.

7. The results of items 4 and 5 provide all the statements of the theorem. We consider an almost-Riemannian structure dened by a Euclidean vector bundle (E, ⟨ , ⟩) and a vector bundle morphism f , and we are interested in local normal forms of orthonormal vector elds dening the structure in a neighborhood of a point p that we can assume to be p = 0 in local coordinates.

These normal forms turn out to be the key of the next section.

First we follow the lines of [START_REF] Agrachev | A Gauss-Bonnet like formula on two-dimensional almost-Riemannian manifolds[END_REF] (also used in [START_REF] Boscain | Normal forms and invariants for 2dimensional almost-Riemannian structures[END_REF] and [START_REF] Boscain | Mason Local properties of almost-Riemannian structures in dimension 3[END_REF]).

Let W be a codimension 1 submanifold transversal to the distribution. We can dene a coordinate system y = (x 2 , . . . , x n ) in W , and choose an orientation transversal to W . Let γ y be the family of normal geodesics parametrized by arclength, transversal to W at y, and positively oriented. The mapping (x 1 , y) → γ y (x 1 ) is a local dieomorphism and the geodesics x 1 → γ y (x 1 ) realize the minimal distance between W = {x 1 = 0} and the surfaces {x 1 = c} for c small enough. The transversality conditions of the PMP are consequently satised along all these surfaces: if λ(x 1 ) is a covector associated to one of these geodesics then the tangent space to {x 1 = c} at γ y (x 1 ) is ker(λ(x 1 )). Now let X 1 = ∂ x 1 be the vector eld dened by X 1 (q) = d dx 1 γ y (x 1 ) at the point q = γ y (x 1 ). It is a unitary vector eld belonging to ∆. Let X 2 , . . . , X n be n-1 vector elds such that {X 1 , X 2 , . . . , X n } be an orthonormal frame of ∆. For geodesics the control functions (u 1 , u 2 , . . . , u n ) from the PMP satisfy u j = ⟨λ, X j ⟩. But here (u 1 , u 2 , . . . , u n ) = (1, 0 . . . , 0) so that ⟨λ, X j ⟩ = 0 for j = 2, . . . , n and the vector elds X 2 , . . . , X n are tangent to the surfaces {x 1 = c}. Consequently the vector elds have the following form:

X 1 =       1 0 . . 0       , X j =       0 a 2,j . . a n,j       for 1 < j < n
for any choice of the coordinates in W and any choice of X 2 , . . . , X n , under the condition that they provide an orthonormal frame related to the sub-Riemannian metric.

Notice that thanks to the transversality conditions the vectors elds X 2 , . . . , X n are not only orthogonal to X 1 for the sub-Riemannian metric but also for the canonical inner product of R n for the chosen coordinates.

Let us assume now that p = 0 belongs to Z r with r ≥ 1 and r 2 ≤ n. We want to show that the coordinates and X 2 , . . . , X n can be chosen in such a way that they write:

X 1 =           1 0 . . . . 0           X 2 =           0 1 + b 2 (x) a 3,2 (x) . . . a n,2 (x)           . . . X n-r =           0 . . 1 + b n-r (x) a n-r+1,n-r . a n,n-r (x)           and X n-r+1 . . X n =         0 . . 0 ------ D(x)         where D(x) ∈ M r (R), b j (0) = a i,j (0)
= 0, and D(0) = 0. Firstly we can assume that X j (0) = ∂ x j for j = 2, . . . , n-r and X j (0) = 0 for j = n -r + 1, . . , n. Indeed X 2 , . . . , X n are tangent to W where we can choose freely the cooordinates (x 2 , . . . , x n ).

If r = n -1, which is generically possible only if n = 2 and r = 1, it is nished. Otherwise we can rst replace X 2 by the normalization of n j=2 a 2j X j . Then we replace X j by X j -a 2j a 22 X 2 for j > 2. These vector elds belong to ∆, their rst two coordinates vanish and they are orthonormal to (the new) X 2 . It remains to orthonormalize these n -2 vector elds. This cannot be done directly because some of them vanish in the singular locus. However they are images by the vector bundle morphism f of locally nonvanishing smooth sections of E that can be orthonormalized. The desired form of the vector elds is obtained by induction.

4.3

Nilpotent and solvable approximations of generic distributions Theorem 7 For a generic distribution holds:

(i) Let p be a tangency point in Z 1 , that is a point where T p Z 1 = ∆ p . Then X n = 0 but X n ̸ = 0, in normal form.

(ii) At all other points, including all points in Z r with r ≥ 2, the nilpotent approximation X 1 , X 2 , . . . , X n is a set of n linearly independant vector elds.

Proof.

Consider a generic distribution, a point p in Z 1 and privileged coordinates centered at p such that

X 1 =         1 0 . . . 0         X 2 =         0 1 + b 2 (x) a 3,2 (x) . . a n,2 (x)         . . . X n-1 =         0 . . 0 1 + b n-1 (x) a n,n-1 (x)         X n =         0 . . . 0 a n (x)         where b j (0) = a i,j (0) = 0. The determinant of X is a n (x)Π n-1 j=2 (1 + b j (x)) and the singular locus is locally Z = Z 1 = {a n = 0}. Let a n (x) = n i=1 α i x i + o(∥x∥).
Since the determinant is a submersion at p there exists i such that α i ̸ = 0.

If there exists i

0 ≤ n -1 such that α i 0 ̸ = 0 then X n ̸ = 0. Moreover da n (0).X i 0 = α i 0 ̸ = 0. Hence X i 0 (0) / ∈ T 0 Z 1 and T 0 Z 1 + ∆ 0 = R n .
2. If α i = 0 for i = 1, . . . , n -1, then X n = 0, but α n ̸ = 0 and X n = (α n x n +q(x 1 , . . . , x n-1 )) ∂ ∂x n , where q(x 1 , . . . , x n-1 ) is quadratic.

Moreover T 0 Z 1 = ker da n (0) = {x n = 0}, hence X j (0) ∈ T 0 Z 1 for j = 1, . . . , n -1 and ∆ 0 = T 0 Z 1 .
Let now p ∈ Z r with r ≥ 2. We can choose privileged coordinates centered at p such that:

X 1 =           1 0 . . . . 0           X 2 =           0 1 + b 2 (x) a 3,2 (x) . . X n-r+1 . . X n =         0 . . 0 ------ D(x)        
where b j (0) = a i,j (0) = 0 and D(x) ∈ M r (R).

Let x = (y, z) where y = (x 1 , . . . , x n-r ) and z = (x n-r+1 , . . . , x n ). The assumption p = 0 ∈ Z r implies D(0) = 0 and we can write:

D(x) = l(y) + k(z) + Q(y, z)
where l and k are linear and Q contains all the terms of degree greater than one. Here l(y) = (l ij (y)) n-r+1≤i,j≤n belongs to M r (R) and each entry l ij is linear w.r.t. y = (x 1 , . . . , x n-r ).

As in the proof of Theorem 6 we denote by d(p) the dimension of the quotient (T p Z r + ∆ p ) /T p Z r . It is clear that here d(0) = dim (dD(0).∆ 0 ), at p = 0. But ∆ 0 = Span{X 1 (0), . . . , X n-r (0)} and for j = 1, . . . , n -r: dD(0).X j (0) = ∂D ∂x j (0) = ∂l ∂x j (0).

Consequently d(0) = rank (l) (as a mapping from R n-r to M r (R)).

For j = n -r + 1, . . . , n the nilpotent approximation X j of X j is

X j = n i=n-r+1 l ij (y) ∂ ∂x i .
This is due to the fact that the weights of the coordinates are 1 for i = 1, . . . , n -r and 2 for i = n -r + 1, . . . , n.

Let us assume that the vector elds X n-r+1 , . . . , X n are not linearly independant. Then the mapping l takes its values in a subspace of M r (R) of dimension r 2 -r and d(p) is smaller than or equal to r 2 -r. But r 2 -r ≤ d(p) ≤ min(n -r, r 2 ) according to the proof of Theorem 6 and the equality can hold for r = 1 only. Indeed let s = min(n -r, r 2 ) -d(p) as in the proof of Theorem 6. Then:

1. If n -r ≥ r 2 then s = r 2 -(r 2 -r) = r. But r ≥ 2 implies s ≥ 2 and s 2 > r which is impossible according to Theorem 6.

2. If n -r < r 2 then s = n -r -(r 2 -r) = n -r 2 . But n, r and s must satisfy n ≥ r 2 + s(r 2 + r + s -n) according to the proof of Theorem 6. For s = n -r 2 this is s ≥ sr which is impossible for r ≥ 2.

■

The canonical coordinates are privileged with weights (1, 2, 5) and the vector elds X 1 , X 2 , and X 3 are homogeneous of order -1 hence equal to their nilpotent approximations. The algebra g is here the ideal of L generated by X 1 that is g = Span{X 1 , X 4 , X 5 , X 6 , X 7 }.

The vector elds X 2 and X 3 are linear, as well as X 8 = 1 2 [X 2 , X 3 ] and

X 9 = [X 2 , X 8 ].
The orders of X 8 and X 9 are respectively -2 and -3 which shows that the vector elds of order smaller than -1 are not necessarily in g.

Notice that the singular locus is here Z = {xy = 0}. ■

Example 3

Consider in R 4 the almost-Riemannian structure dened by the vector elds:

X 1 = ∂ x , X 2 = ∂ y + x∂ z , X 3 = y∂ w , X 4 = 1 2 x 2 ∂ z + 1 2 y 2 ∂ w .
Since [X 1 , X 2 ] = ∂ z and [X 2 , X 3 ] = ∂ w , the coordinates (x, y, z, w) are privileged with weights (1, 1, 2, 2) at p = (0, 0, 0, 0). The vector elds X 1 , X 2 are homogeneous of order -1 and independent at 0, and the vector eld X 3 is homogeneous of order -1 but vanishes at 0. The last eld X 4 is homogeneous of order 0. Consequently the rst three are equal to their nilpotent approximation and X 4 = X 4 . According to the notations of Section 3 we have k = 2 and m = 3.

The Lie algebra L is spanned by X 1 , X 2 , X 3 , X 4 and

X 5 = [X 1 , X 2 ] = ∂ z , X 6 = [X 2 , X 3 ] = ∂ w , X 7 = [X 1 , X 4 ] = x∂ z .
Despite the fact that X 3 (0) = X 3 (0) = 0 we cannot assert as in the nilpotent case that X 3 does not belong to g (see Section 3.3 after Theorem 3). Indeed the ideal generated in L by X 1 and X 2 is here

g = Span{X 1 , X 2 , X 3 , X 5 , X 6 , X 7 } because X 3 = [X 2 , X 4 ].
As explained in Section 3.3 this may happen when k < m < n.

Notice that the determinant of X 1 , X 2 , X 3 , X 4 is -1 2 x 2 y. Therefore the singular locus is Z = {xy = 0} which shows that the structure is not generic.

4 . 2

 42 Normal forms of almost-Riemannian structures

Examples

The following four examples use the notations of Section 3.

The rst one is a standard example of a solvable approximation on the group Heisenberg. Example 2 shows that the elements of the Lie algebra L of nonholonomic order smaller than -1 are not necessarily in g. We exhibit a vector eld of the distribution that vanishes at p but belongs to the ideal g in Example 3 (which implies that the rank of the nilpotent approximation is not full).

To nish the Lie algebra L of Example 4 is not solvable, it contains a semi-simple subalgebra. Recall that this is not generic.

In these four examples the vector elds are equal to their nilpotent or solvable approximations at 0. It is of course possible to add terms of higher nonholonomic order without modifying the conclusions.

Example 1

Consider in R 3 the almost-Riemannian structure dened by the vector elds:

At p = (0, 0, 0) the coordinates (x, y, z) are privileged with weights (1, 1, 2), the vector elds X 1 and X 2 are homogeneous of order -1 and X 3 is homogeneous of order 0, so that X 1 = X 1 , X 2 = X 2 , X 3 = 0 and X 3 = X 3 . The nilpotent approximation at p is not an almost-Riemannian structure, it is the constant rank 2 sub-Riemannian structure dened by X 1 and X 2 . The Lie algebra generated by

here the Heisenberg Lie algebra, and X 3 is a linear vector eld on g. Finally X 1 , X 2 , X 3 is a linear ARS on the Heisenberg group.

Example 2

The almost-Riemannian structure is here dened in R 3 by:

The Lie algebra L contains X 1 , X 2 , X 3 and

■

In the general case the Lie algebra L need not be solvable. Indeed it is a subalgebra of the semi-direct product of g by its algebra of derivations. But the algebra of derivations of a nilpotent Lie algebra is not solvable in general. For instance the derivations of the Heisenberg algebra is the set of endomorphisms the matrix of which writes in the canonical basis:

The subalgebra of such derivations that moreover satisfy e = f = a + d = 0 is equal to sl 2 hence semisimple.

Example 4 illustrates that possibility.

Example 4

Consider in R 5 , with coordinates (

The coordinates (x, y, z, w, t) are privileged with weights (1, 1, 1, 2, 2) at the origin. At this point the vector elds X 1 , X 2 , X 3 (resp. X 4 , X 5 ) are homogeneous of order -1 (resp. 0), hence equal to their nilpotent approximations (resp. X 4 = X 4 and X 5 = X 5 ).

Since X 10 = [X 4 , X 5 ], [X 10 , X 4 ] = 2X 4 and [X 10 , X 5 ] = -2X 5 the vector elds X 4 , X 5 and X 10 , that do not belong to g, generate a semi-simple Lie algebra isomorphic to sl 2 . Consequently the algebra L is not solvable.

The singular locus is here Z = {xyz = 0}, and again the structure is not generic.

■


Kiswendsida Abel Ouedraogo
email: kiswendsidaabel.ouedraogo@univ-valenciennes.fr

Simon Enjalbert
email: simon.enjalbert@univ-valenciennes.fr

Frédéric Vanderhaegen
email: frederic.vanderhaegen@univ-valenciennes.fr

How to Learn from the Resilience of Human-Machine Systems?

Keywords: Human-Machine Systems, Resilience, BCD model, Learning process, Feedback/Feedforward control

In this paper, we aim to analyse the resilience of Human-Machine Systems (HMS) in order to improve it from learning process. A State of Art is achieved and resilience engineering of HMS is defined. Then, human-machines' learning processes supposed to improve systems' resilience and indicators proposed in the literature to assess it are analysed. A perspective can be to propose an efficient indicator, for instance based on Benefit-Cost-Deficit (BCD) model, which can lead to the system resilience characterisation.

INTRODUCTION

A Human-Machine System (HMS) is a system where the human operators and machines cooperate to ensure an optimal operation. In the literature, many studies have focused on the machine system only, not on the response capacity of the whole HMS which is regularly subject to external and/or internal disturbances. In this paper, we analyse resilient systems, i.e. HMS able to cope with a disruption in its response capabilities, and how to learn from them.

The article is organized as follows. In the second section, a State of the Art on the concept of resilience is given. Some descriptions of resilient HMS are presented. A few existing formulations to assess the potential resilience of systems are given in the third section; we also introduce a BCD model applied to analyse human activities on a rail simulation. The forth section describes some structures in order to characterise and enhance HMS resilience. Finally, we give the conclusion and perspectives.

STATE OF THE ART ON RESILIENCE

Resilience is a relatively new field of research although the concept has been first use in physics for Charpy impact test in 1905 [START_REF] Jacobs | Engineering Materials Technology[END_REF][START_REF] Charpy | Note sur l'essai des métaux à la flexion par choc de barreaux entaillés, Mémoire et compterendus de la Société des ingénieurs civils de France[END_REF]. Resilience was related to the ability of a material to recover from a shock or disturbance.

The concept of resilience was next developed in the field of ecology and characterises natural systems that tend to maintain their integrity when subject to disturbances [START_REF] Ludwig | Sustainability, stability and resilience[END_REF]. It has generated much interest in different communities and has been applied to: psychology, psychiatry [START_REF] Goussé | Apport de la génétique dans les études sur la résilience : l'exemple de l'autisme[END_REF], sociology, economy, biology [START_REF] Orwin | New indices for quantifying the resistance and resilience of soil biota to exogenous disturbances[END_REF][START_REF] Pérez-España | An inverse relationship between stability and maturity in models of aquatic ecosystems[END_REF]Sánchez, 2001), informatics (Chen et al., 2007;[START_REF] Nakayama | Fault-resilient sensing in wireless sensor networks[END_REF]Luo and Yang, 2007), automatic [START_REF] Numanoglu | Energy efficiency and error resilience in coordinated and non-coordinated medium access control protocols[END_REF].

In psychology, the concept is linked to the invulnerability theory i.e. the positive capacity of people to cope with trauma and to bounce back.

In biology, resilience is developed in the theory of viability i.e. ability for an organism to survive after disruption.

In industrial systems, resilience is related to the concept of robustness [START_REF] Amalberti | Violations et migrations ordinaires dans les activités à risques: conséquences pour la résilience globale et la gestion du retour d'expériences en entreprise[END_REF] which is related to errorresistant and error-tolerant systems.

In organisational and safety management, resilience is the capacity of a system to survive, adapt and grow face to unforeseen changes, even catastrophic incidents [START_REF] Zieba | Resilience and affordances: perspectives for humanrobot cooperation[END_REF].

In HMS, the definition from [START_REF] Wreathall | Properties of resilient organisations: An initial view[END_REF] can be adopted: "Resilience is the ability of an organization (system) to maintain, or recover quickly to, a stable state allowing it to continue operations during and after a major mishap or in presence of a continuous stress". That approach incorporated all components of the system (machines, human, and organisation) and their interactions.

The Fig. 1 illustrates some issues: when can we say that HMS is resilient and why? If not, what to do to make it happen? What can we learn from this HMS resilience? In this paper, we focus on defining the HMS resilience and so, how to learn from it. Fig. 1. Pending issues about resilient HMS.

Resilient HMS?

What to do to make it so?

Learning Process (BCD model)

No Yes

Resilient systems are supposed to adapt to unplanned events by its capacities to anticipate failures, to control disturbances, to react and to recover from unexpected events. These systems organisation is illustrated in Fig. 2.

The system has also the possibility to learn from its reactions to unplanned events. The resilient HMS must have some characteristics that authors described as resilience's components. Fig. 3 illustrates three qualities or resilience components (knowledge, competence and resources) that a system must have to be able to remain in control, and therefore to be resilient. [START_REF] Hollnagel | Epilogue : Resilience engineering precepts[END_REF]. The system must constantly update its knowledge, competence and resources by learning from successes and failures.

A non resilient system cannot continue to operate after a major mishap or in presence of continuous stress. Then how can it be done? Several mechanisms can be established for instance:

-Using non-affected elements to compensate and achieve the functions of parts affected [START_REF] Lin | Adaptive error-resilience transcoding using prioritized intrarefresh for video multicast over wireless networks[END_REF][START_REF] Nakayama | Fault-resilient sensing in wireless sensor networks[END_REF][START_REF] Numanoglu | Energy efficiency and error resilience in coordinated and non-coordinated medium access control protocols[END_REF]. What works compensates what does not; -Maintaining the system between minimum and maximum thresholds (acceptability domain) rather than a stability value [START_REF] Martin | La résilience dans les modèles de systèmes écologiques et sociaux[END_REF]; -Putting critical elements in redundancy (Luo and Yang, 2007). The affected elements are no longer solicited and are replaced by redundant components; -Learning or re-learning the affected elements to work correctly or to work better [START_REF] Cheveau | Approche pratique de la culture de sécurité : pour une maîtrise des risques industriels plus efficace[END_REF]. What does not work is reset and prepared for a future operation; -Learning initial functions for Human i.e. the ability to develop attitudes or behaviours ensuring the survival of the organism [START_REF] Marcantoni | Mécanismes cellulaires de l'apprentissage[END_REF].

RESILIENCE AND MEASUREMENTS

A first measurement of resilience, related to the ability of a material to recover from a shock, has been given in Charpy impact test. [START_REF] Hollnagel | Epilogue : Resilience engineering precepts[END_REF] argue that resilience itself cannot be measured; only the potential for resilience can be. There is currently no objective measurement for assessing the resilience of a HMS but several different methods have been proposed in the literature.

3.1 Measurements [START_REF] Martin | La résilience dans les modèles de systèmes écologiques et sociaux[END_REF] evaluates the resilience by maximum intensity of an absorbable force by the system without perturbing its functioning. For [START_REF] Luo | Enabling fault resilience for web services[END_REF], the measurement of the instantaneous resilience is linked with the speed of recovery from a disturbance. Both of them do not consider the total time of disturbance which is an important requirement according to the definition of resilience. [START_REF] Orwin | New indices for quantifying the resistance and resilience of soil biota to exogenous disturbances[END_REF]) link resilience with the measurement of the instantaneous and maximal disturbance. (Pérez-España and Arreguín-Sánchez, 2001) calculate the resilience by the opposite of the tangent of the result of ratio between resistance and the recovery time of a disturbance. These measurements are related to a minimum acceptable performance threshold based and defined by designers and/or users of the system. As disturbances can lead to change system functioning goals, measurements cannot be assessed this way over the time.

We propose to define the principles of benefit-Cost-Deficit model [START_REF] Vanderhaegen | The benefit-cost-deficit (BCD) model for human analysis and control[END_REF][START_REF] Vanderhaegen | A reinforced iterative formalism to learn from human errors and uncertainty[END_REF] that could be used as an approach to characterise the HMS resilience.

BCD model principles

The BCD model is based on indicators that assess the consequences of deviated human behaviours on several criteria related to technical or human performances or states. Positive consequences are Benefits and negative ones are acceptable Costs when the undesirable events are under control or Deficits when they are over control.

The logical value of functions B, C and D for an evaluation criterion when it is subjective or qualitative can be turned into a numeric value or an objective function by the following expression (1):

       = - = otherwise b a D if b a K b a C if b a K b a B if b a K b a K t a s t b s b a K i i D i i C i i B i J a i b i i J 0 ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( )) ( ( )) ( ( ) , ( , , , , , (1)
Where [START_REF] Gu | Humanmachine systems resilience -Safety application[END_REF]:

-a and b are situations at times t a and t b respectively, -s i (x(t x )) is the severity criterion associated to the position x at time t x .

Application to the COR&GEST platform

The COR&GEST platform is a railway simulation platform developed in the University of Valenciennes. It involves a miniature railway structure on which several trains can move. The platform has a supervision interface to manage remotely signals and devices, and a driving interface for each train, both illustrates in Fig. 4. Trains transmit videos from miniature cameras to the driving interface. So, human operators can drive the miniature trains using several indicators such as the train speed, the speed limit and signals that are virtually integrated into the view recorded from the cameras. The traffic is managed by the supervision interface completed by a scenario working station that allow building schedules and managing technical failure occurrences.

In order to study the human behaviour during the train driving activities with or without any technical failure occurrence, an experimental protocol was built with 7 inexperienced human operators. After a training phase for driving familiarisation on the platform, the human operators have to control a set of scenarios during 2 hours for two separates operational conditions:

-Without any disturbance, i.e. in normal conditions, to study the capacity of the operator to maintain the stability of the system in routine driving conditions; -With disturbances to study respect of rules and procedures by human operators after an unplanned or sudden disturbance occurrence such as a break default, a temporary speed limitation, etc.

Then several performance criteria are assessed:

-The traffic safety in terms of barriers non violation, i.e. signals and speed limits respect, -The departure quality related to the respect of trains departure time from stations, -The arrival quality related to the respect of trains arrival time at stations, -The human workload linked to the number of interactions between drivers and technical driving support systems.

The traffic safety criterion is considered as a potential deficit, benefits and costs are related to others criteria. Results, illustrated in Table 1, allow concluding that monotonous and repetitive tasks during a long time period can lead to a system performance decrement regarding BCD model parameters without disturbance. Furthermore, the failure occurrence can [START_REF] Vanderhaegen | Rail simulation to study human reliability[END_REF]:

-Cause a breakdown in monotonous activity, -Increase the situation awareness, -Improve the control of several performance or safety criteria.

Learning Process

A resilient HMS must be able to react and recover from unplanned events. In order to enhance his "resilience", the system has to carry out learning process [START_REF] Vanderhaegen | Pronostic, diagnostic et tâtonnement face à des situations dynamiques[END_REF] from reactions (successes and failures) through different methods such as:

-Observation of behaviour or actions, -Imitation or copy of response's actions to an event [START_REF] Calinon | On Learning, Representing, and Generalizing a task in a Humanoid Robot[END_REF], -Groping or Trial-and-Error testing, -Intuition, without reasoning.

Machine learning, inspired by related psychological theory, is the process of acquiring knowledge, skills and attitudes by modelling (Observation, Imitation) or by action (Groping, Intuition). It improves the system performance within resources and skills available [START_REF] Buche | Apprentissage par imitation pour les comportements d'agents autonomes[END_REF].

In engineering context, we are interested in the learning processes which are classified into two approaches. Supervised learning, illustrated in output Process Action controller ---------------guidance -errors learner Fig. 5, where the supervised learner is "led" by a controller which provides guidance on actions to be performed in order to improve its performances. In the opposite, an unsupervised learner must discover by himself. Reinforcement learning, illustrated in Fig. 6, where the reinforcement learning agent dynamically interacts with the environment. With evaluative feedback passing back from the environment, a value function is formulated to guide the action towards the desired goal [START_REF] Quah | Reinforcement learning combinedwith a fuzzy adaptive learning control network (FALCON-R) for pattern classification[END_REF]. Fig. 6. Reinforcement learning (adapted from [START_REF] Kazuo | Learning scheduling control knowledge through reinforcements[END_REF].

For a HMS involved into an unexpected or unplanned disruption, reinforcement learning appears to be more adapted. Under uncertainty, the system interacts with environment to search for optimal decision-making behaviour. Thus reinforcement learning pursues a twofold objective:

-Lead optimally a system under pressure (optimal control in automatics), -Learn or re-learn the optimal control through experiments (trial-error's method).

Feedback/Feedforward control model

In order to be in control, a system must necessary know what has happened, e.g. with feedback control, what happens, by diagnostics, and what may happen, e.g. with feedforward control. The objective is to provide the required qualities, illustrated previously in Fig. 3, for the HMS. A great number of research works have proposed feedback and/or feedforward controllers using different methods in order to reach the mentioned objectives. [START_REF] Lee | Iterative learning control of permanent magnet linear motor with relay automatic tuning[END_REF] have proposed a feedback/feedforward control structure, illustrated in Fig. 7, for the trajectory tracking of a linear DC (Direct Current) motor. The control model uses a conventional PID (Proportional Integral Derivative) feedback control to stabilise the system and a feedforward control using an iterative learning control (ILC) algorithm to enhance the tracking trajectories performances by capitalising on the experience gathered from the repeated execution of the same operations.

PID controller is the most common general-purpose controller using a feedback control mechanism. The proportional term handles the present state of the system, the integral term handles its past, and the derivative term tries to predict and handle the future. Feedforward refers to perception or anticipation; so feedforward control can be linked to learned anticipatory responses to known causes. Fig. 7. Block diagram of Iterative Learning Control (from [START_REF] Lee | Iterative learning control of permanent magnet linear motor with relay automatic tuning[END_REF]. [START_REF] Jang | Iterative Learning Control in Feedback Systems[END_REF] proposed an ILC model through feedback/feedforward structure, illustrated in Fig. 8, for sharp tracking control of a manipulator robot. Feedback configuration ensures learning and feedforward is used for prediction. Feedforward controller is based on the rules updated by previous trial from the plant.

Therefore, the Feedback-Feedforward control model aims to provide the required qualities, illustrated in Fig. 3, for HMS resilience.

Behaviour in response to unexpected events

Facing unexpected events, a resilient HMS can react and overcome an issue applying one of the two behavioural approaches illustrated in Fig. 9.

In the first approach, we can first consider the current state can be identified by experience when system faces a known situation. Thus, a prognosis on the future states of the system can be determined e.g. by feedforward control, which is to assess or predict the future states based on the current state and various parameters of HMS. If the current state cannot be identify, in case of unforeseen event, or a prognosis not possible, a diagnosis will be performed e.g. by feedback control related to the previous states of the system in order to assess current one.

To sum up, if the current state of the HMS can be identify or assess by diagnosis, then a prognosis related to the reachable future states is done in order to select appropriate alternative or reaction to recover from disruption. Resilience on known situations is then characterised.

The second approach focuses on the occurrence of a new or unprecedented situation. In this case, identification, prognosis and diagnosis are not possible. A trial and error action without knowing consequences on the HMS, like in reinforcement learning, must be realised. Actions can be performed iteratively [START_REF] Vanderhaegen | A reinforced iterative formalism to learn from human errors and uncertainty[END_REF] for repetitive systems tasks, by experience or through a feedback/feedforward control model. Resilience improvement under unforeseen situations can be studied considering this approach.

The purpose of the algorithm illustrated in Fig. 9 is to achieve the selection of the most appropriate alternative and to define a new action plan with its associated consequences that will be applied to HMS in order to evaluate its impact in terms of resilience.

CONCLUSIONS

In this paper, we propose to analyse the resilience of Human-Machine Systems (HMS) in order to assess the potential resilience of systems and to improve it. A State of the Art on resilience and an adopted definition for HMS are given. Then, we presented some structures and methods for the improvement of system resilience. We also give a few indicators in literature used to measure system resilience. The BCD model is presented and applied on the COR&GEST rail simulation platform to analyse human activities.

In future works, we may use BCD model to define an efficient indicator for the assessment of HMS resilience.

Fig. 3 .

 3 Fig. 3. Required qualities of a resilient system (from Hollnagel and Woods, 2006). These qualities must be exercised continuously. Knowledge is needed in order to recognise what to expect or where to focus next (anticipation). The competence refers to know what to look for or where to pay attention. The resources refer to the ability to know what to do and how to do it[START_REF] Hollnagel | Epilogue : Resilience engineering precepts[END_REF]. The system must constantly update its knowledge, competence and resources by learning from successes and failures.

Fig. 4 .

 4 Fig. 4. The supervision and driving interfaces of the COR&GEST platform.

  Fig. 5. Supervised learning (adapted from Buche, 2002).

Fig. 8 .

 8 Fig. 8. Learning model with feedback/feedforward structure (from Jang et al, 1995).

  Fig. 9. Behaviour in response to unexpected events (adapted from Vanderhaegen, 2009a).

Table 1 .

 1 Quantitative assessment of the BCD model parameters (negative values: costs or deficits; positive values: benefits).

			without disturbance			with disturbance
			Departure	Arrival	Interaction		Departure	Arrival	Interaction
		safety	delay	delay	number	safety	delay	delay	number
	Human operator 1	-406	379	-103	-948	-401	-10	-161	-1058
	Human operator 2	-24	397	-83	-666	-328	196	-168	-818
	Human operator 3	-653	-160	-70	-902	-76	450	-179	-861
	Human operator 4	-168	340	-260	-391	-423	740	-325	-688
	Human operator 5	-167	71	-76	-674	-185	273	-159	-527
	Human operator 6	-610	407	-114	-800	-176	290	-216	-867
	Human operator 7	-21	257	-80	-724	-157	192	173	-775
	Average	-293	242	-112	-729	-249	305	-148	-799