Sub-Riemannian Geometry and Optimal Transport

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Sub-Riemannian Geometry and Optimal Transport

Ludovic Rifford October 1, 2012

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Preface

The main goal of these lectures is to give an introduction to sub-Riemannian geometry and optimal transport, and to present some of the recent progress in these two fields. This set of notes is divided into three chapters and two appen- dices. Chapter 1 is concerned with the notions of totally nonholonomic distributions and sub-Riemannian structures. The concepts of End-Point mappings and singular horizontal paths which play a major role through these lectures are introduced here. Chapter 2 deals with sub-Riemannian geodesics. We study first and second-order variations of the End-Point mapping to derive necessary and sufficient conditions for an horizontal path to be minimizing. We provide several examples, including the Montgomery counter-example of singular minimizing curve. In Chapter 3, we study the Monge problem for sub-Riemannian quadratic costs. We give a crash-course in optimal transport theory and ex- plain how the sub-TWIST condition together with the Lipschitz regularity of a ”variational” cost implies the well-posedness of Monge’s problem. Then we study the fine regularity properties of sub-Riemannian distances to obtain existence and uniqueness of optimal transport maps in the sub-Riemannian context. We recall basic facts on ordinary differential equations in Appendix 1 and less classical results of differential calculus in normed vector spaces in Appendix 2. The latter plays a key role in Chapter 2.

The reader of these notes should be familiar with the basics in differential geometry and measure theory. Possible references in these fields include the textbooks by Lee [Lee03] and Evans-Gariepy [EG92]. For further reading, we strongly encourage the reader to look at other texts in sub-Riemannian geometry and optimal transport. Multiple viewpoints always lead to deeper understanding and may open new directions for research. Among them, we may suggest the textbooks by Montgomery [Mon02], Agrachev, Barilari and Boscain [ABB12], and Villani [Vil08].

This set of notes grew from a series of lectures that I gave during a CIMPA school in Beyrouth, Lebanon, on the invitation of Fernand Pelletier. I take the opportunity of this preface to warmly thank Ali Fardoun, Mohamad Mehdi and Fernand Pelletier who organized the school, Ahmed El Soufi for his support and friendship, and through him the ”Centre International de Math´ematiques Pures et Appliqu´ees”. My gratitude goes also to all faculties and students who attended this sub-Riemannian CIMPA school in making it a success.

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Sub-Riemannian structures

Throughout all the chapter,M denotes a smooth connected manifold without boundary of dimensionn≥2.

1.1 Totally nonholonomic distributions

Distributions

A smoothdistribution∆ of rankm≤n(m≥1) onM is a rankmsubbundle of the tangent bundle T M, that is a smooth map that assigns to each point xofM a linear subspace ∆(x) of the tangent spaceTxM of dimensionm. In other terms, for every x∈M, there are an open neighborhood V^x ofxin M andmsmooth vector fieldsX_x¹,· · · , X_x^mlinearly independent onV^xsuch that

∆(y) = Spann

X_x¹(y),· · ·, X_x^m(y)o

∀y∈ V^x.

Such a family of smooth vector fields is called alocal frameinV^xfor the distribution ∆. All the distributions which will be considered later will be smooth with constant rankm∈[1, n]. Thus, from now on, ”distribution” always means

”smooth distribution with constant rank”. A co-rankkdistribution onM is a distribution of rankm=n−kand any smooth vector fieldX onM such that X(x)∈∆(x) for anyx∈M is called a section of ∆.

Example 1.1.1. We call trivial distribution on M the rank n distribution

∆ defined by ∆(x) = TxM for all x ∈ M. For topological reasons, such a distribution may not admit non-vanishing sections (for example, by the hairy ball theorem, there is no non-vanishing continuous vector fields on any even dimensional sphere).

Example 1.1.2. InR³ with coordinates(x, y, z), the distribution∆ defined by

∆(x, y, z) =Spann

X(x, y, z), Y(x, y, z)o

∀(x, y, z)∈R³ with

X =∂x−y

2∂z and Y =∂y+x 2∂z, is a rank 2(or co-rank 1) distribution on R³.

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Example 1.1.3. More generally, if x= (x1, . . . , xn, y1, . . . , yn, z)denotes the coordinates in R²ⁿ⁺¹ and the 2n smooth vector fields X¹, . . . , Xⁿ, Y¹, . . . , Yⁿ are defined by

Xⁱ =∂xi−yi

2∂z, Yⁱ =∂yi+xi

2∂z ∀i= 1, . . . , n, then the distribution∆ defined by

∆(x) =Spann

X¹(x), , . . . , Xⁿ(x), Y¹(x), . . . , Yⁿ(x)o

∀x∈R²ⁿ⁺¹, is a co-rank1 distribution onR²ⁿ⁺¹.

Example 1.1.4. Let α be a smooth non-degenerate 1-form on M, that is a 1-form which does not vanish (αx 6= 0 for any x ∈ M). The distribution ∆ defined as

∆(x) =Ker(αx) ∀x∈M, is a co-rank1 distribution onM.

Example 1.1.5. As an example, consider the unit 3-sphere S³ in R⁴ with coordinates (x1, y1, x2, y2), that is

S³=n

(x1, y1, x2, y2)∈R⁴|x²₁+y²₁+x²₂+y₂²= 1o . Let αbe the smooth non-degenerate1-form on S³ defined by

α=

x1dy1−y1dx1+x2dy2−y2dx2

|_S3

, then∆ =Ker(α)is a co-rank1 distribution onS³.

We say that a given distribution ∆ onM admits aglobal frame if there are msmooth vector fieldsX¹,· · ·, X^monM such that

∆(x) = Spann

X¹(x),· · ·, X^m(x)o

∀x∈M.

In general, distributions do not admit global frames (see Example 1.1.1). It is worth noticing that in the particular case ofRⁿ all distributions are trivial.

Proposition 1.1.6. Any distribution inRⁿ admits a global frame.

Proof. Let us first show how to construct a non-vanishing section of a given distribution inRⁿ.

Lemma 1.1.7. Let ∆ be a distribution of rank m in Rⁿ. Then there is a non-vanishing smooth vector fieldX such that X(x)∈∆(x), for anyx∈Rⁿ. Proof of Lemma 1.1.7. Define the multivalued mappingδ:Rⁿ→2^Rⁿ by

δ(x) =n

v∈∆(v)| |v|= 1o

∀x∈Rⁿ.

By construction, δ is locally Lipschitz with respect to the Hausdorff distance on compact subsets of Rⁿ. By compactness of ¯B(0n,2), there is ∈ (0,1)

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1.1. TOTALLY NONHOLONOMIC DISTRIBUTIONS 3 such that for any x, y ∈ B(0¯ n,2) with |x−y| < , and any v ∈ δ(x), there is w ∈ δ(y) such that |v−w| < 1. Let N ≥ 2 be an integer such that the increasing sequence of ballsB¹, . . . ,B^N defined by

Bⁱ=B(0n, i) ∀i= 1, . . . , N,

satisfies ¯B(0n,1)⊂ B^N. For everyx∈Rⁿ, we denote by Proj_δ(x)the projection onto the (m−1)- dimensional sphere δ(x). Note that the mapping Proj_δ(x)is well-defined and ”smooth” on the open set

O^x=n

w∈Rⁿ| hv, wi 6= 0,∀v∈δ(x)o .

For everyi∈ {1, . . . , N−1}, consider a smooth mappingPi :Bⁱ⁺¹ → Bⁱ such that

|Pi(x)−x|< ∀x∈ Bⁱ⁺¹. (1.1) Note that such a smooth function exists becauseBⁱ is a ball and Bⁱ⁺¹ is contained in the-neighborhood ofBⁱ. Let ¯w∈δ(0) be fixed. We define the vector fieldX : ¯B(0n,1)→Rⁿ as follows:

We first set

X1(x) = Proj_δ(x)( ¯w) ∀x∈ B¹. Then, givenXi :Bⁱ→Rⁿ, we defineXi+1:Bⁱ⁺¹→Rⁿ as

Xi+1(x) = Proj_δ(x)

Xi Pi(x)

∀x∈ Bⁱ⁺¹. By construction (by (1.1) and the definition of ), Xi Pi(x)

belongs to O^x for anyx∈ Bⁱ⁺¹. In conclusion, X =XN is smooth on ¯B(0n,1) and satisfies 0n 6=X(x) ∈ δ(x) for any x ∈ B(0n,1). Repeating the construction on the annuli B(0n,2)\B(0n,1), B(0n,3)\B(0n,2), . . ., we obtain a non-vanishing section of ∆ onRⁿ.

We now prove Proposition 1.1.6 by induction onm. Let ∆ be a rank (m+1) distribution onRⁿ. By Lemma 1.1.7, it admits a non-vanishing sectionX on Rⁿ. The multivalued mapping ˜∆ :Rⁿ→2^Rⁿ defined by

∆(x) = ∆(x)˜ ∩n

X(x)o⊥

∀x∈Rⁿ,

is a smooth rank m distribution (here {X(x)}^⊥ denotes the space which is orthogonal to X(x) with respect to the Euclidean scalar product). Thus by induction, there are smooth vector fieldsX¹, . . . , X^monRⁿ such that

∆(x) = Span˜ n

X¹(x), . . . , X^m(x)o

∀x∈Rⁿ. The family{X¹, . . . , X^m, X} is a global frame for ∆.

A finite family of smooth vector fields{X¹, . . . , X^k} is called agenerating family for ∆ onM if there holds

∆(x) = Spann

X¹(x),· · · , X^k(x)o

∀x∈M.

Any distribution can be represented by a generating family.

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Proposition 1.1.8. Let∆ be a distribution of rankm≤nonM. Then there are k=m(n+ 1) smooth vector fields X¹,· · ·, X^k such that {X¹,· · ·, X^k} is a generating family for ∆.

Proof. By definition, for everyx∈M, there is an open neighborhoodV^x ofx inM andmsmooth vector fieldsX_x¹,· · · , X_x^mlinearly independent onV^xsuch that

∆(y) = Spann

X_x¹(y),· · ·, X_x^m(y)o

∀y∈ V^x.

Since M is paracompact, there is a locally finite coveringV ={Vⁱ}ⁱ∈I where each open setVⁱ equals V^xi for somexi∈M.

Lemma 1.1.9. There are a locally finite open covering {U^j}^j∈J of M and a partition∪ⁿ⁺¹l=1Jl ofJ such that the following properties are satisfied:

(a) for every j∈J, there isi=i(j)∈I such that U^j ⊂ Vⁱ, (b) for everyl∈ {1, . . . , n+ 1}and any j6=j⁰∈Jl,U^j∩ U^j⁰=∅.

Proof of Lemma 1.1.9. Recall that every smooth manifold is triangulable. Let T ={T^t}^t∈T be a triangulation ofM that refines the covering{Vⁱ}ⁱ∈I, in the sense that the closure of each face F of T is a subset of some Vⁱ. For every α∈ {0, . . . , n}, denote byT^α={Tt^α}^t∈Tα the family ofα-dimensional faces in T. For everyα∈ {0, . . . , n},we can construct easily a collection of open sets W^α={Ws^α}^s∈Sα satisfying the following properties:

- W^α is a refinement of{Vⁱ}ⁱ∈I, - ∪^t∈TαTt^α⊂ ∪^s∈SαWs^α,

- eachWs^α is an open neighborhood of someα-dimensional face ofT^α, - for anys6=s⁰ ∈Sα,Ws^α∩ Ws^α⁰ =∅,

- for anys6=s⁰ ∈S0,Ws^α∩ Ws^α⁰ =∅,

- for anyα∈ {1, . . . , n}and anys6=s⁰ ∈Sα,Ws^α∩ Ws^α⁰ ⊂ ∪^t∈Tα−1Tt^α⁻¹. For that, it suffices to proceed by induction on α and to make use of the properties of a triangulation. We conclude easily.

Let us now show how to construct for every r ∈ {1, . . . , m} a family of sections {X₁^j, . . . , X_n+1^j |1 ≤ j ≤ r} of ∆ such that Span{X_l^j(x)|1 ≤ j ≤ r,1≤l≤n+ 1} has dimension≥r for anyx∈M. We proceed by induction onr.

First, for each l ∈ {1, . . . , n+ 1} and each j ∈Jl, there is i =i(j) ∈I such that U^j ⊂ Vⁱ = V^xi. Modifying X_i¹ = X_x¹_i outside U^j if necessary, we may assume thatX_i¹ is defined onM, does not vanish onU^j, and vanishes outside U^j. DefineX₁¹, . . . , X_n+1¹ by

X_l¹= X

j∈Jl

X_i(j)¹ ∀l= 1, . . . , n+ 1.

By construction (Lemma 1.1.9 (b)), the interior of the supports of theX_i(j)¹ ’s are always disjoint. Therefore, each X_l¹ is a non-vanishing section of ∆ on

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1.1. TOTALLY NONHOLONOMIC DISTRIBUTIONS 5

∪j∈JlU^j. This shows that Span{X_l¹(x)|1≤l≤n+ 1}has dimension≥1 for anyx∈M.

Assume now that we have constructed a family of smooth vector fields{X_i^j,|1≤ j≤r,1≤i≤n+ 1}such that

Spann

X_l^j(x)|1≤j≤r,1≤l≤n+ 1o

has dimension ≥ r for any x∈ M (with r < m). For every j ∈ J, there is s=s(j)∈ {1, . . . , m}such that

Spann

X_x^s_i_(j)(x), X_l^j(x)|1≤j≤r,1≤l≤n+ 1o has dimension≥r+ 1 for anyx∈ U^j. DefineX₁^r+1, . . . , X_n+1^r+1 by

X_l^r+1= X

j∈Jl

X_i(j)^s(j) ∀l= 1, . . . , n+ 1.

We leave the reader to check that by construction (modifying the X_x^s(j)

i(j)’s if necessary as above), the vector space

Spann

X_l^j(x)|1≤j ≤r+ 1,1≤l≤n+ 1o has dimension≥r+ 1 for anyx∈M. The proof is complete.

The H¨ormander condition

Recall that for any smooth vector fieldsX, Y onM given by X(x) =

n

X

i=1

ai(x)∂x_i, Y(x) =

n

X

i=1

bi(x)∂x_i,

in local coordinates x = (x1, . . . , xn), the Lie bracket [X, Y] is the smooth vector field defined as

[X, Y](x) =

n

X

i=1

ci(x)∂xi, wherec1, . . . , cn are the smooth scalar function given by

ci=

n

X

j=1

∂xjbi

aj− ∂xjai

bj ∀i= 1,· · ·, n.

For the upcoming controllability results (like the Chow-Rashesvky Theorem), it is important to keep in mind the following dynamical characterization of the Lie bracket.

Proposition 1.1.10. LetX, Y be two smooth vector fields in an neighborhood ofx∈Rⁿ. Then we have

[X, Y](x) := DxY ·X(x)−DxX·Y(x)

= lim

t→0

e⁻^tY ◦e⁻^tX ◦e^tY ◦e^tX(x)−x

t² , (1.2)

wheree^tX ande^tY denote respectively the flows ofX andY.

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bx

b

e^tX(x)

be^tY ◦e^tX(x)

b

e⁻^tX◦e^tY ◦e^tX(x)

b e^−tY ◦e^−tX◦e^tY ◦e^tX(x)

Proof. All the functions appearing in the proof will be defined locally fortclose to 0 and/or in a neighborhood ofx. Define the smooth functionh4by

h4(t) := e⁻^tY ◦e⁻^tX◦e^tY ◦e^tX(x) ∀t.

We haveh⁰₄(0) = 0. As a matter of fact, we have for anyt, h⁰₄(t) =−Y(h4(t)) + ∂

∂xe⁻^tY

(t,h3(t))

·h⁰₃(t) whereh3 is defined byh3(t) := e^−tX ◦e^tY ◦e^tX(x). Then we have

h⁰₃(t) =−X(h3(t)) + ∂

∂xe^−tX

(t,h2(t))

·h⁰₂(t), whereh2(t) := e^tY ◦e^tX

(x) and h⁰₂(t) =Y(h2(t)) +

∂

∂xe^tY

(t,h1(t))

·h⁰₁(t),

with h1(t) := e^tX(x) and h⁰₁(t) = X(e^tX(x)). Since partial derivatives of the form _∂x^∂ e^tX at t = 0 are equal to Id, we get h⁰₁(0) = X(x), h⁰₂(0) = X(x) +Y(x), h⁰₃(0) = Y(x) and h⁰₄(0) = 0. Therefore, the left-hand side of (1.2) is equal to ¹₂h⁰⁰₄(0). By derivating the above formulas, we get

h⁰⁰₁(0) =dX(h1(0))·h⁰₁(0) =dX(x)·X(x), and

h⁰⁰₂(0) =dY(h2(0))·h⁰₂(0) +

"

d dt

"

∂

∂xe^tY

(t,h₁(t))·h⁰₁(t)

##

t=0

. ButdY(h2(0))·h⁰₂(0) =dY(x)·(X(x) +Y(x)) and

"

d dt

"

∂

∂xe^tY

(t,h1(t))

·h⁰₁(t)

##

t=0

=

"

d dt

∂

∂xe^tY

(t,h1(t))

#

t=0

·h⁰₁(0) + ∂

∂xe^tY

(0,h1(0))

·h⁰⁰₁(0)

=

"

∂²

∂t∂x e^tY

(0,x)

+ ∂²

∂x² e^tY

(0,x)·h⁰₁(0)

#

·X(x) +dX(x)·X(x)

= ∂

∂x ∂

∂te^tY

(0,x)

·X(x) +dX(x)·X(x)

=dY(x)·X(x) +dX(x)·X(x).

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1.1. TOTALLY NONHOLONOMIC DISTRIBUTIONS 7 We infer that h⁰⁰₂(0) = dY(x)·(2X(x) +Y(x)) +dX(x)·X(x). In the same way, we have

h⁰⁰₃(0) =−dX(h3(0))·h⁰₃(0) +

"

d dt

"

∂

∂xe^−tX

(t,h2(t))

·h⁰₂(t)

##

t=0

,

−dX(h3(0))·h⁰₃(0) =−dX(x)·Y(x) and

"

d dt

"

∂

∂xe⁻^tX

(t,h2(t))

·h⁰₂(t)

##

t=0

=

"

d dt

∂

∂xe⁻^tX

(t,h2(t))

#

t=0

·h⁰₂(0) + ∂

∂xe⁻^tX

(0,h2(0))

·h⁰⁰₂(0)

=−dX(x)·(X(x) +Y(x)) +dY(x)·(2X(x) +Y(x)) +dX(x)·X(x)

=−dX(x)·Y(x) +dY(x)·(2X(x) +Y(x)).

Which impliesh⁰⁰₃(0) =−2dX(x)·Y(x) +dY(x)·(2X(x) +Y(x)). Finally h⁰⁰₄(0) = −dY(h4(0))·h⁰₄(0) +

"

d dt

"

∂

∂xe^−tY

(t,h3(t))

·h⁰₃(t)

##

t=0

=

"

d dt

∂

∂xe⁻^tY

(t,h₃(t))

#

t=0

·h⁰₃(0) + ∂

∂xe⁻^tY

(0,h₃(0))·h⁰⁰₃(0)

= −dY(x)·Y(x)−2dX(x)·Y(x) +dY(x)·(2X(x) +Y(x))

= 2(dY(x)·X(x)−dX(x)·Y(x))

= 2[X, Y](x), which concludes the proof.

Remark 1.1.11. We check easily that the following properties are satisfied:

(i) Given smooth vector fieldsX1, X2, Y1, Y2 anda1, a2∈R, we have [a1X1+a2X2, Y1] = a1[X1, Y1] +a2[X2, Y1]

[X1, a1Y1+a2Y2] = a1[X1, Y1] +a2[X1, Y2].

(ii) Given smooth vector fields X andY, we have[X, Y] =−[Y, X].

(iii) Given three smooth vector fieldsX, Y, Z, the Jacobi identity is satisfied:

X,[Y, Z]+

Y,[Z, X]+

Z,[X, Y]= 0.

Remark 1.1.12. Given a smooth diffeomorphism φ from a smooth manifold U to a smooth manifold V andX a smooth vector field on V , we recall that the push-forwardφ_∗(X)of X is defined by

φ_∗(X)(y) :=Dφ⁻¹(y)φ X(φ⁻¹(y)

∀y∈ V. We have

[φ_∗(X), φ_∗(Y)] =φ_∗([X, Y]).

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For any family F of smooth vector fields on an open set O ⊂ M, we denote by Lie(F) the Lie algebra of vector fields generated by F. It is the smallest vector subspace S of X^∞(M) (the space of smooth vector fields on M) containing F that also satisfies

[X, Y]∈S ∀X ∈ F, ∀Y ∈S.

It can be constructed as follows: Denote by Lie¹(F) the space spanned byF inX^∞(M) and define recursively the spaces Lie^k(F) (k= 1,2, . . .) by

Lie^k+1(F) = Span

Lie^k(F)∪n

[X, Y]|X ∈ F, Y ∈Lie^k(F)o

∀k≥0.

This defines an increasing sequence of vector spaces inX^∞(M) satisfying Lie(F) = [

k≥1

Lie^k(F).

In general, Lie(F) is an infinite-dimensional subspace ofX^∞(M).

Example 1.1.13. Let Abe an×nreal matrix, bbe a vector inRⁿ, andX, Y be the smooth vector fields in Rⁿ defined by

X(x) =Ax, Y(x) =b ∀x∈Rⁿ.

The non-zero Lie brackets of X andY are always constant vector fields of the form

ad¹_X(Y) := [X, Y] =−Ab, ad²_X(Y) :=

X,ad¹_X(Y)=A²b, and

ad^k+1_X (Y) :=h

X,ad^k_X(Y)i

= (−1)^k+1A^k+1b ∀k≥0.

By the Cayley-Hamilton Theorem,Aⁿ can be expressed as a linear combination ofA⁰, . . . , Aⁿ⁻¹. Therefore, Lie(X, Y)is the set of vector fieldsZ inRⁿ of the form

Z(x) =λAx+

n−1

X

i=0

λiAⁱb ∀x∈Rⁿ, with λ, λ0, . . . , λ_n−1∈R. It is a finite-dimensional Lie algebra.

Example 1.1.14. Let X, Y be the two smooth vector fields inR² (with coordinates x= (x1, x2)) defined by

X(x) =∂x1, Y(x) =f(x1)∂x2 ∀x∈R²,

where f is a smooth scalar function. Then, Lie(X, Y) is the space of smooth vector fields spanned byX and

ad^k_Y(X) =f^(k)∂x2 fork≥0.

Thus, Lie(X, Y) is infinite-dimensional whenever the derivatives off span an infinite-dimensional space of functions.

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1.1. TOTALLY NONHOLONOMIC DISTRIBUTIONS 9 For any pointx∈M, Lie(F)(x) denotes the set of all tangent vectorsX(x) withX ∈Lie(F). It follows that Lie(F)(x) is always a linear subspace ofTxM, hence finite-dimensional.

Example 1.1.15. Returning to Example 1.1.14 and denoting by (e1, e2) the canonical basis ofR², we check that

Lie(X, Y)(x) =Spann

e1, f^(k)(x1)e2|k= 0,1,2, . . .o

∀x∈R². In particular, Lie(X, Y)(x) = Re1 if f(x) and all its derivatives at x vanish and Lie(X, Y)(x) =R² otherwise.

We say that the smooth vector fields X¹, . . . , X^m satisfy theH¨ormander condition on some open setO ⊂M if and only if

Lien

X¹,· · · , X^mo

(x) =TxM ∀x∈ O.

A distribution ∆ onM is calledtotally nonholonomiconM if for everyx∈M, there are an open neighborhoodV^xofxinM and a local frameX_x¹,· · ·, X_x^mon V^xwhich satisfies the H¨ormander condition on V^x. This definition is intrinsic, it does not depend upon the choice of the local frameX_x¹, . . . , X_x^m. This is a consequence of the following result:

Proposition 1.1.16. Let {X¹, . . . , X^m},{Y¹, . . . , Y^m} be two families of linearly independent smooth vector fields on an open setO ⊂M such that

Spann

X¹(x), . . . , X^m(x)o

=Spann

Y¹(x), . . . , Y^m(x)o

∀x∈ O. Then there holds for any integerk≥1,

Lie^kn

X¹, . . . , X^mo

(x) =Lie^kn

Y¹, . . . , Y^mo

(x) ∀x∈ O.

Proof. It is sufficient to show that the following inclusion holds for any integer k≥2,

Lie^kn

X¹, . . . , X^mo

(x)⊂Lie^kn

Y¹, . . . , Y^mo

(x) ∀x∈ O.

Since the Y^j(x) are always linearly independent, there are smooth functions α^j_i :O →Rwithi, j= 1, . . . , m,such that

Xⁱ(x) =

m

X

j=1

α^j_i(x)Y^j(x) ∀x∈ O,∀i= 1, . . . , m.

Then for everyi= 1, . . . , mand every smooth vector fieldZ, there holds [Xⁱ, Z] =





m

X

j=1

α^j_iY^j, Z



=

m

X

j=1

α^j_i[Y^j, Z]−

m

X

j=1

dα^j_i(Z)Y^j. Since Span

X¹(x), . . . , X^m(x) ⊂ Span

Y¹(x), . . . , Y^m(x) for any x, this shows that

Lie²n

X¹, . . . , X^mo

(x)⊂Lie²n

Y¹, . . . , Y^mo

(x) ∀x∈ O. We conclude easily by an inductive argument.

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We also observe that any generating family for ∆ does satisfy the H¨ormander condition provided ∆ is totally nonholonomic.

Proposition 1.1.17. Let ∆ be a totally nonholonomic distribution on M and{X¹, . . . , X^k} be a generating family for ∆. Then X¹, . . . , X^k satisfy the H¨ormander condition onM.

Proof. We need to show that Lien

X¹,· · ·, X^ko

(x) =TxM ∀x∈M.

Letx∈M be fixed. By assumption, there is an open neighborhoodV^x and a local frameY_x¹,· · · , Y_x^monV^x which satisfies the H¨ormander condition onV^x. Proceeding as in the proof of Proposition 1.1.16, we show that

Lie^kn

X¹, . . . , X^ko

(x)⊂Lie^kn

Y_x¹, . . . , Y_x^mo (x),

for every integerk ≥ 1. This proves that X¹, . . . , X^k satisfy the H¨ormander condition onM.

Remark 1.1.18. Since for any smooth vector fieldX, there holds[X, X] = 0, a one dimensional distribution cannot be totally nonholonomic.

Degree of nonholonomy

If ∆ is a rankmtotally nonholonomic distribution onM, then for everyx∈M, there are an open neighborhoodV^xofxandmsmooth vector fieldsX_x¹, . . . , X_x^m which satisfy the H¨ormander condition onV^x. We calldegree of nonholonomy of ∆ atxthe smallest integerr=r(x)≥1 such that

Lie^rn

X¹, . . . , X^mo

(x) =TxM.

Thanks to Proposition 1.1.16, this definition does not depend upon the choice of the local frame. Moreoever, we shall say that ∆ is totally nonholonomic of degreerif the nonholonomy degree of any point inM is≤r.

Example 1.1.19. The distribution given in Example 1.1.2 is totally nonholonomic. We check easily that

[X, Y] =∂z ∀i, j= 1, . . . , n, which means that∆ has degree2.

Example 1.1.20. More generally, the distribution given in Example 1.1.3 is totally nonholonomic of degree 2. We check easily that

[Xⁱ, Y^j] =δij∂z ∀i, j= 1, . . . , n.

Example 1.1.21. The Martinet distribution inR³ (with coordinates(x, y, z)) is defined as

∆(x, y, z) =Spann

X(x, y, z), Y(x, y, z)o

∀x∈R³,

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1.1. TOTALLY NONHOLONOMIC DISTRIBUTIONS 11 where

X=∂x, Y =∂y+x² 2 ∂z. The first Lie bracket ofX, Y is given by

[X, Y] =x∂z. For any(x, y, z)∈R³ with x6= 0, the three vectors

X(x, y, z), Y(x, y, z),[X, Y](x, y, z)

are linearly independent. Hence, ∆ is a totally nonholonomic distribution of degree2 on R³\ {x= 0}. The Lie bracket[[X, Y], Y]is given by

[[X, Y], Y] =∂z.

Then,∆ is a totally nonholonomic distribution of degree 3 onR³. Example 1.1.22. More generally, ifX, Y are given by

X =∂x, Y =∂y+x^l∂z,

with l ∈ N^∗, we check easily that the distribution spanned by X and Y is a totally nonholonomic distribution of degreel+ 1.

Example 1.1.23. Assume that M has dimensionn = 2p+ 1 and let α be a 1-form on M satisfying

α∧(dα)^p6= 0

then the distribution given by∆ =Ker(α)is totally nonholonomic of degree2.

Such a1-form is called a contact form and the associated distribution is called a contact distribution. As a matter of fact, given x¯ ∈M, there is a local set of coordinates(x1, . . . , xn)in an open neighborhood V¯ ofx¯ such thatαhas the form

α=

2p

X

i=1

aidxi

! +dxn,

wherea1, . . . , a2p are smooth scalar function onV¯ such that ai(¯x) = 0 ∀i= 1, . . . ,2p.

Hence, the family of smooth vector fieldsX¯¹, . . . ,X¯^2p given by X¯ⁱ=∂x_i−ai∂x_n ∀i= 1, . . . ,2p,

defines a local frame for∆ =Ker(α)in V¯. On the one hand, the n= 2p+ 1- formα∧(dα)^p atx¯ reads

(α∧(dα)^p)_x_¯= X

σ∈P2p



 Y

l=1,...,p

∂ajl

∂xil

−∂ail

∂xjl



dxn∧(dxi1∧dxj1). . .∧ dxip∧dxjp

|^x^¯, (1.3)

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whereP^2pdenotes the set ofp-tuples of the formσ= ((i1, j1), . . . ,(ip, jp))with {i1, j1, . . . , ip, jp} ={1, . . . ,2p} and il < jl for all l = 1, . . . , p. On the other hand, we check easily that

X¯ⁱ,X¯^j(¯x) = ∂xiaj−∂xjai

∂xn(¯x) ∀i, j= 1, . . . ,2p.

Therefore, if there is¯i∈ {1, . . . ,2p} such that[ ¯X^¯ⁱ,X¯^j](¯x) = 0 for anyj 6=i, then all the products appearing in (1.3) vanish, which implies that(α∧(dα)^p)_x_¯= 0, contradiction. We deduce that for everyi∈ {1, . . . , n}, there holds

Spann

X¯¹(¯x), . . . ,X¯^2p(¯x),X¯ⁱ,X¯¹(¯x), . . . ,X¯ⁱ,X¯^2p(¯x)o

=T¯xM. (1.4) This means that ∆ =Ker(α)is totally nonholonomic of degree 2.

Example 1.1.24. As an example, the 1-form given in Example 1.1.5 is a contact form onS³. There holds

α∧dα = (x1dy1−y1dx1+x2dy2−y2dx2)∧(2dx1∧dy1+ 2dx2∧dy2)

= 2x1dy1∧dx2∧dy2−2y1dx1∧dx2∧dy2

+2x2dx1∧dy1∧dy2−2y2dx1∧dy1∧dx2. A basis of the tangent space to S³ at x = (x1, y1, x2, y2) ∈ S³ is given by (V1, V2, V3) with







V1 = −y1e1+x1e2−y2e3+x2e4

V2 = −x2e1+y2e2+x1e3−y1e4

V3 = −y2e1−x2e2+y1e3+x1e4. Then

(α∧dα)_x(V1, V2, V3) =

2x²₁ x²₁+y²₁+x²₂+y₂²

−2y₁² −x²₁−y₁²−x²₂−y²₂ + 2x²₂ x²₁+y²₁+x²₂+y₂²

−2y₂² −x²₁−y₁²−x²₂−y²₂

= 2 x²₁+y₁²+x²₂+y₂²²= 2.

This means that the restriction of the 3-form α∧dα to the tangents spaces to S³ does not vanish.

1.2 Horizontal paths and End-Point mappings

Horizontal paths

Let ∆ be a distribution of rankm≤ninM. A continuous pathγ: [0, T]→Rⁿ is said to be horizontal with respect to ∆ if it is absolutely continuous with square integrable derivative (see Appendix A) and satisfies

˙

γ(t)∈∆ γ(t)

a.e. t∈[0, T].

For every x∈ M and every T > 0, we denote by Ω^x,T_∆ the set of horizontal paths γ: [0, T] →M starting atx. If ∆ admits a global frameX¹, . . . , X^m, then there is a one-to-one correspondence between Ω^x,T_∆ and an open subset of L²([0, T];R^m).

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1.2. HORIZONTAL PATHS AND END-POINT MAPPINGS 13 Proposition 1.2.1. LetF=

X¹, . . . , X^m be a global frame for∆. Then for every x∈M and every T >0, there is an open subset U_F^x,T ofL²([0, T];R^m) such that the mapping

u∈U_F^x,T 7−→γu∈Ω^x,T_∆ ,

(whereγu: [0, T]→M is the unique solution to the Cauchy problem

˙ γu(t) =

m

X

i=1

ui(t)Xⁱ(γu(t)) a.e. t∈[0, T], γu(0) =x,) (1.5) is one-to-one.

Proof. The set of controlsu∈L²([0, T];R^m) such that the solutionγuof (1.5) is well-defined on [0, T] is a non-empty open set. Moreover, by construction, any path γu is absolutely continuous with square integrable derivative and almost everywhere tangent to ∆. This proves that the map under study is well-defined. Letγ∈Ω∆,x,T be such that there areu, v∈L²([0, T];R^m) such that

˙ γ(t) =

m

X

i=1

ui(t)Xⁱ(γ(t)) =

m

X

i=1

vi(t)Xⁱ(γ(t)) a.e. t∈[0, T].

Since the tangent vectors X¹(γ(t)), . . . , X^m(γ(t)) are always linearly independent inTγ(t)M, we infer thatu(t) =v(t) for almost everyt∈[0, T], which proves that our map is injective. Furthermore, givenγ∈Ω^x,T_∆ , for almost every t∈[0, T], the pathγis differentiable attand there is a uniqueu(t)∈R^msuch that ˙γ(t) =Pm

i=1ui(t)Xⁱ(γ(t)). By construction, the functionu: [0, T]→R^m belongs toL²([0, T];R^m).

As seen before, a general distribution may have no global frame, but it can be represented byk=m(n+ 1) vector fields (see Proposition 1.1.8).

Proposition 1.2.2. Let F =

X¹,· · · , X^k be a generating family for∆ on M. Then, for everyx∈M and every T >0, there is an open subset U_F^x,T of L² [0, T];R^k

such that the mapping

u∈U_F^x,T 7−→γu∈Ω^x,T_∆ ,

(whereγu: [0, T]→M is the unique solution to the Cauchy problem

˙ γu(t) =

k

X

i=1

ui(t)Xⁱ(γu(t)) a.e. t∈[0, T], γu(0) =x,) (1.6) is onto.

Proof. Letγ ∈Ω^x,T_∆ be fixed. For everyt ∈[0, T], there is an open setO^tof γ(t) inM andmintegersi^t₁, . . . , i^t_m∈ {1, . . . , k}such that

Span n

Xⁱ^t¹(x), . . . , Xⁱ^t^m(x)o

= ∆(x) ∀x∈ O^t.

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The curveγ([0, T]) is compact and is contained in∪^t∈[0,T]O^t. Hence, there are N timest1, . . . , tN ∈[0, T] together with a partition of unity{ψj} such that

[0, T]⊂

N

[

j=1

O^tj, Supp (ψj)⊂ O^tj,

N

X

j=1

ψj = 1.

For everyj, there is a smooth mappingUj:T M →R^m such that

v=

m

X

l=1

Uj(v)Xⁱ^tj^l (x),

for every (x, v)∈T M withx∈ O^tj andv∈∆(x). Then, there holds for almost every t∈[0, T] and any j∈ {1, . . . , N},

γ(t)∈ O^tj =⇒ γ(t) =˙

m

X

l=1

Uj( ˙γ(t))Xⁱ^tj^l γ(t) . By the properties satisfied by{ψj}, we infer that

˙ γ(t) =

N

X

j=1

ψj γ(t)

"_m X

l=1

Uj( ˙γ(t))Xⁱ^tj^l γ(t)

#

=

N

X

j=1 m

X

l=1

ψj γ(t)

Uj( ˙γ(t))

Xⁱ^tj^l γ(t) , for almost every t ∈ [0, T]. Each mapping t 7→ ψj γ(t)

Uj( ˙γ(t)) belongs to L² [0, T];R. We infer easily the existence of u ∈ L² [0, T];R^k such that γ=γu.

Remark 1.2.3. If M is compact, then solutions to (1.5) (resp. (1.6)) are defined for anyu∈L²([0, T];R^m) (resp. u∈L²([0, T];R^k)).

Given a family of smooth vector fields F =

X¹,· · ·, X^k on M and x ∈ M, T > 0, a function u ∈ U_F^x,T ⊂L² [0, T];R^k is called a control and the corresponding solution of (1.6) is called the trajectory starting at x and associated with the control u. Since any horizontal path can be viewed as a trajectory associated to a control system like (1.6), we restrict in the next paragraph our attention to End-Point mappings associated with finite families of smooth vector fields.

End-Point mappings LetF =

X¹, . . . , X^k be a family ofk≥1 smooth vector fields onM. As before, givenxandT >0, there is a maximal open subsetU_F^x,T ⊂L² [0, T];R^k such that for everyu∈U_F^x,T, there is a unique solution to the Cauchy problem

˙ γu(t) =

k

X

i=1

ui(t)Xⁱ(γu(t)) a.e. t∈[0, T], γu(0) =x. (1.7)

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1.2. HORIZONTAL PATHS AND END-POINT MAPPINGS 15 The End-Point mapping associated to F at x in time T > 0 is defined as follows,

E_F^x,T : U_F^x,T −→ M u 7−→ γu(T).

Givenu∈U_F^x,T, we denote by X_F^u the time-dependent vector field defined by X_F^u(t, x) :=

m

X

i=1

ui(t)Xⁱ(x) a.e. t∈[0, T],∀x∈M.

Its flow Φ^u_F(t, x) is well-defined and smooth on a neighbourhood ofx; we denote by DxΦ^u_F(t, x) its differential at (t, x) with respect to the x variable. The following result holds. (We refer the reader to Appendix A for reminders in differential equations and to Appendix B for reminders in differential calculus in infinite dimension.)

Proposition 1.2.4. The End-Point mappingE_F^x,T is of classC¹ onU_F^x,T and for every controlu∈U_F^x,T, its differentiable atu,

DuE_F^x,T : L²([0, T];R^k)−→T_E^x,T

F (u)M

is given by

DuE_F^x,T(v) =DxΦ^u_F(T, x)· Z T

0

DxΦ^u_F(t, x)−1

·X_F^v t, E_F^x,t(u)

dt (1.8) for everyv∈L²([0, T];R^k). Moreover, the mapping

u∈U_F^x,T 7−→ DuE_F^x,T (1.9)

is locally Lipschitz.

Proof. Any smooth manifold can be smoothly embedded in an Euclidean space.

Then without loss of generality we can assume thatM is a smooth submanifold of some R^N and consequently that the Xⁱ’s are the restrictions of smooth vector fields ˜X¹, . . . ,X˜^k which are defined in an open neighborhood ofM in R^N. Givenu∈U_F^x,T andv∈L²([0, T];R^k) let us look at

lim→0

1

E_F^x,T u+v

−E_F^x,T u . Using the previous notations, we have

γu+v(T) = Z T 0

k

X

i=1

(ui(t) +vi(t))Xⁱ γu+v(t) dt

= Z T 0

k

X

i=1

(ui(t) +vi(t)) ˜Xⁱ γu+v(t)

dt, (1.10) with γu+v(0) = x. For every i = 1, . . . , k and every t ∈ [0, T], the Taylor expansion of each ˜Xⁱ at γu(t) gives

X˜ⁱ γu+v(t)

= ˜Xⁱ γu(t)

+Dγu(t)X˜ⁱ· γu+v(t)−γu(t)

+|γu+v(t)−γu(t)|o(1). (1.11)

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Setting δx(t) :=γu+v(t)−γu(t) for anyt, we may assume thatδx has size, then (1.10) yields formally

δx(T) = Z T 0

k

X

i=1

ui(t)Dγu(t)X˜ⁱ · δx(t)dt +

m

X

i=1

vi(t) ˜Xⁱ γu(t) + o().

This suggests that the function t ∈ [0, T] 7→ δx(t) should be solution to the Cauchy problem

δ˙x(t) =

" _k X

i=1

ui(t)D_γ_u_(t)X˜ⁱ

# δx(t) +

" _k X

i=1

vi(t) ˜Xⁱ γu(t)

#

a.e. t∈[0, T], (1.12) with δx(0) = 0. By (1.10)-(1.12) together with Gronwall’s Lemma (see Ap- pendix A) we check easily that for everyv∈L² [0, T];R^k, the quantity

1

E_F^x,T u+v

−E_F^x,T u

−δx(T)

tends to zero as tends to zero. For almost every t∈[0, T], denote by Au(t) the matrix in MN(R) representing the linear operator Pk

i=1ui(t)Dγu(t)X˜ⁱ in the canonical basis ofR^N and for everyt∈[0, T], denote byBu(t) the matrix in MN,k(R) whose the columns are the ˜Xⁱ(γu(t))’s. Denote bySu : [0, T] → MN(R) the solution to the Cauchy problem

S˙u(t) =Au(t)Su(t) a.e. t∈[0, T], Su(0) =In. Note thatSu(t) is exactly the Jacobian of the flow Φ^u_˜

F(with ˜F ={X˜¹, . . . ,X˜^k}) at (t, γu(t)) with respect to thexvariable. The solution of (1.12) at timeT is given by (see Appendix A)

δx(T) =DuE_F^x,T(v) =Su(T)Z T 0

Su(t)⁻¹Bu(t)v(t)dt.

Thus we check that (1.8) is satisfied. Let us now prove the local Lipschitzness ofu7→DuE_F^x,T and indeed give more details on the estimates that were needed in the above proof. Let ¯ua control be fixed inU_F^x,T ⊂L²([0, T];R^k). The curve γu¯([0, T])⊂M ⊂R^N is compact. Let >0 be fixed, the setV ⊂R^N defined by

V :=n

γu¯(t) +z|t∈[0, T], z∈B(0, )o

is relatively compact. Then there isK >0 such that all the ˜Xⁱ’s are bounded byK onV and all the ˜Xⁱ’s areK-Lipschitz onV. Set

δ:=

KT e^KT^k^u^¯^k^L²

and pick a control u∈ L²([0, T];R^k) with ku−u¯kL² < δ. We claim that u belongs to U_F^x,T and that the trajectory γu : [0, T] → M ⊂ R^N (which is

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1.2. HORIZONTAL PATHS AND END-POINT MAPPINGS 17 associated withu) is contained inV. Argue by contradiction and assume that there is ¯t∈[0, T] such thatγu(t) is on the boundary ofV. Taking ¯t >0 smaller if necessary, we may assume thatγu(t) belongs toV for anyt∈[0,¯t). Set

f(t) :=|γu(t)−γu¯(t)| ∀t∈[0,¯t].

Then we have for everyt∈[0,¯t), f(t) =

Z t 0

k

X

i=1

ui(s) ˜Xⁱ γu(s)

−

k

X

i=1

¯

ui(s) ˜Xⁱ γu¯(s) ds

≤ Z t

0

k

X

i=1

(ui(s)−u¯i(s)) ˜Xⁱ γu(s)

ds +Z t

0

k

X

i=1

u¯i(s)

X˜ⁱ γu(s)

−X˜ⁱ γu¯(s)

ds

≤ Kt u−u¯

_L₂+Z t 0

k

X

i=1

¯ ui(s)

Kf(s)ds.

By Gronwall’s Lemma (see Appendix A) and definition ofδ, we infer that f ¯t

≤KT u−u¯

L²e^KT^k^u^¯^k^L² < .

Thus we get a contradiction and the claimed is proved. Letu, u⁰ ∈L²([0, T];R^k) withku−u¯kL²,ku⁰−u¯kL² < δ, by repeating the same argument we get

|γu⁰(t)−γu(t)| ≤Kt u⁰−u

L²e^KT(ku¯kL2+δ) ∀t∈[0, T].

(This shows that End-Point mappings are locally Lipschitz.) Denote bySu, Su⁰ : [0, T]→MN(R) the solutions to the Cauchy problems

S˙u(t) =Au(t)Su(t) a.e. t∈[0, T], Su(0) =In, S˙u⁰(t) =Au⁰(t)Su⁰(t) a.e. t∈[0, T], Su⁰(0) =In, whereAu, Au⁰ are defined by

Au(t) :=

k

X

i=1

ui(t)JX˜ⁱ γu(t)

, Au(t) :=

k

X

i=1

u⁰_i(t)JX˜ⁱ γu⁰(t) ,

for almost everyt∈[0, T] (JX˜ⁱ(γu(t)) (resp. JX˜ⁱ(γu⁰(t)) denotes the Jacobian matrix of ˜Xⁱ at γu(t) (resp. at γu⁰(t))). Taking K > 0 larger if necessary, we may assume that it is an upper bound for theJX˜ⁱ’s onV and a Lipschitz constant for theJX˜ⁱ’s onV. Then we have for everyt∈[0, T],

kSu(t)k =

In+Z t 0

Au(s)Su(s)ds

≤ 1 +Z t 0

k

X

i=1

|ui(s)|

JX˜ⁱ γu(t)

kSu(s)k ds

≤ 1 +Z t 0

K

k

X

i=1

|ui(s)| kSu(s)kds.

Sub-Riemannian Geometry and Optimal Transport