soft open fences

# LOCAL NON-INJECTIVITY OF THE EXPONENTIAL MAP AT CRITICAL POINTS IN SUB-
RIEMANNIAN GEOMETRY

Samuël Borza<EMAIL_ADDRESS>Wilhelm Klingenberg<EMAIL_ADDRESS>
###### Abstract

We prove that the sub-Riemannian exponential map is not injective in any
neighbourhood of certain critical points. Namely that it does not behave like
the injective map of reals given by $f(x)=x^{3}$ near its critical point
$x=0$. As a consequence, we characterise conjugate points in ideal sub-
Riemannian manifolds in terms of the metric structure of the space. The proof
uses the Hilbert invariant integral of the associated variational problem.

Keywords— Sub-Riemannian geometry, Conjugate points, Metric geometry
MSC (2020)— 53C17, 58E10, 53C22

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Non-local injectivity of the sub-Riemannian exponential map
    1. 3.1 A family of normal extremals and the associated flow
    2. 3.2 Hilbert invariant integrals
    3. 3.3 The sub-Riemannian conjugate locus
    4. 3.4 Non-local injectivity of the sub-Riemannian exponential map
  4. 4 Applications and final remarks
    1. 4.1 Metric geometry
    2. 4.2 Structure of the sub-Riemannian cut locus
    3. 4.3 Final remarks

##  1 Introduction

In their 1932 landmark paper, Morse and Littauer [ML32] showed that the
exponential map of an analytic Finsler manifold is never injective on any
neighbourhood of a conjugate vector. In other words, conjugate points occur
precisely when certain families of extremals (called extremal fields) fail to
cover the neighbourhood of these points in a one-to-one manner. Savage [Sav43]
extended this result to the smooth case, while Warner [War65] supplied a
different proof of the same result by obtaining normal forms of the
exponential map near those conjugate vectors at which Whitney’s singularity
theory can be applied.

Our results in the area of sub-Riemannian geometry also require fine
properties of regularity and continuity of the exponential map as established
in a previous work of the present authors [BK21]. This allows us to define
regular/singular conjugate points. As the sub-Riemannian exponential map
corresponds to the projection of solutions of a Hamiltonian system on the
cotangent bundle, geodesic variations may vanish of infinite order in this
regime, at least in the non-analytic case. This is in marked contrast to
Riemannian geometry where the Jacobi differential equation has order two. We
therefore introduce the notion of order of a conjugate points and make a
distinction between those of infinite and finite order. Furthermore, the
proofs of Warner, Savage, and Morse and Littauer all depend on some regularity
of the conjugate locus, which remains open in sub-Riemannian geometry and that
we also address. Our first main result is the following.

######  Theorem 1.

Let $M$ be a sub-Riemannian manifold and $p\in M$. If
$\lambda_{0}\in\mathrm{Dom}(\mathrm{exp}_{p})\subseteq\mathrm{T}^{*}_{p}(M)$
is a strongly normal and regular conjugate covector of finite order, then the
exponential map $\mathrm{exp}_{p}$ is not injective in any neighbourhood of
$\lambda_{0}$.

The content of Theorem 1 is a local extension of the following global known
fact in (sub-)Riemannian geometry: if the structure does not admit abnormal
extremals, and if $\gamma$ is a length-minimizing geodesic with conjugate
endpoints $p=\gamma(0)$ and $q=\gamma(1)$, then on any neighborhood
$V\subseteq M$ of $q$ the exponential map
$\mathrm{exp}_{p}|_{\mathrm{exp}_{p}^{-1}(V)}:\mathrm{exp}_{p}^{-1}(V)\to M$
is not injective (see [ABB20, Theorem 8.73. and Corollary 8.74.]). Thus such a
result is not local, but rather global, and related with minimality: the pairs
of geodesics joining $p$ with some of the points $q\in V$ can a priori be far
away at intermediate times. We do not think that the proof of the local result
can be reduced to the global one, except in very specific cases like in
Section 4.2, we prove the following from simple arguments (see Theorem 39).

######  Theorem 2.

Let $M$ be an ideal sub-Riemannian manifold, and $p\in M$. If
$\lambda_{0}\in\mathrm{Cut}(p)\setminus\mathrm{Cut}^{1}(p)$, then the
exponential map $\mathrm{exp}_{p}$ fails to be injective in any neighbourhood
of $\lambda_{0}\in\mathrm{T}^{*}_{p}(M).$

In the statement above, $\mathrm{Cut}(p)$ denotes the cotangent cut locus,
i.e. the set of initial covectors corresponding to geodesics that are
minimising up to time $t=1$ but not minimising up to time $t=1+\varepsilon$,
for any $\varepsilon>0$. The subset $\mathrm{Cut}^{1}(p)$ of $\mathrm{Cut}(p)$
consists of the covectors $\lambda_{0}$ for which there exists another
$\lambda_{0}^{\prime}\in\mathrm{Cut}(p)$ such that
$\mathrm{exp}_{p}(\lambda_{0})=\mathrm{exp}_{p}(\lambda_{0}^{\prime})$.

Already in Riemannian geometry, it does not seem possible to obtain the local
result as expressed in Theorem 1 in full generality just as a variation on the
theme (see Section 4.3). These two results appear completely different nature
for at least one reason: the proof of the non-local injectivity of the
exponential map relies heavily on the regularity of the conjugate locus in the
neighbourhood of a critical point. Indeed, in the neighbourhood of a generic
critical point, the conjugate locus is a smooth hypersurface. We obtain the
sub-Riemannian analogous of this result in Theorem 29, which is of independent
interest. Because of the Hamiltonian structure of geodesics in sub-Riemannian
geometry, we must introduce a technical notion of order finiteness of a
critical point. This assumption is not necessary for the global result but we
do not know if it is possible to prove Theorem 29 without it.

Let us describe the main ideas of the proof, which uses a variational
argument, in the spirit of Morse and Littauer [ML32], and Savage [Sav43].
Assume that the family of normal extremals starting at $p$ covers the
conjugate point in a one-to-one manner. This premise allows to introduce two
versions of the Poincaré-Cartan one-form: the smooth and exact one-form
$\eta^{*}_{p}$ on the space of initial conditions, and $\eta_{p}$ on the
manifold itself which is only continuous due to the presence of a conjugate
point. Both forms induce line integrals $I^{*}$ and $I$ respectively,
reminiscent of the Hilbert invariant integral in the classical theory of the
calculus of variations. The line integral $I^{*}$ is path-independent and we
then claim that $I$ is also path-independent. This fact is shown by proving
that the sub-Riemannian conjugate locus is a hypersurface in the neighbourhood
of a regular singularity of finite order, and by using an approximation
argument which is available because of Sard’s theorem. From the invariance of
$I^{*}$, we finally conclude that the geodesic joining $p$ to the conjugate
point must be length-minimising, reaching a contradiction.

Theorem 1 contributes to the general theory of conjugate points in sub-
Riemannian geometry but also advances in metric geometry. In [SS09] Shankar
and Sormani introduced a priori different synthetic notions of conjugate
points, i.e. definitions that only depend on the metric structure of the
space. Our main result implies some equivalence between them in the ideal
case, that is to say when non-trivial abnormal geodesics are absent,
demonstrating that if discrepancies are to be found in sub-Riemannian
geometry, abnormal segments or infinite order of conjugate points must play a
role.

######  Theorem 3.

Let $M$ be an ideal sub-Riemannian manifold, $\gamma:\interval{0}{1}\to M$ be
a normal geodesic such that its initial covector $\lambda_{0}$ is regular
conjugate and has finite order, and denote $p:=\gamma(0)$ and $q:=\gamma(1)$.
Then, the following statements are equivalent:

  1. (i)

$q$ is conjugate to $p$ along $\gamma$;

  2. (ii)

$q$ is one-sided conjugate to $p$ along $\gamma$;

  3. (iii)

$q$ is symmetrically conjugate to $p$ along $\gamma$.

Furthermore, if $p$ and $q$ are unreachable conjugate points along $\gamma$,
then $q$ is also conjugate to $p$ along $\gamma$.

The paper is organised as follows. In Section 2, we summarise notions from
sub-Riemannian geometry that will be important in our argument. The family of
extremals is introduced in Section 3.1 while in Section 3.2 we define the
Hilbert integrals. In Section 3.3, we address the regularity of the sub-
Riemannian conjugate locus. This result is a contribution to sub-Riemannian
geometry of independent interest. Finally, we prove Theorem 1 in Section 3.4
and discuss Theorem 3, as well as possible future work and related open
questions, in Section 4.

## Data availability statement

Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

## Acknowledgements

We would like to thank Andrei Agrachev and Luca Rizzi for their interest and
comments.

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No. 945655).

##  2 Preliminaries

We begin with a general description of sub-Riemannian geometry, one that
includes rank-varying structures. We refer the reader to [ABB20], for example,
for details on sub-Riemannian geometry.

A sub-Riemannian structure on a manifold $M$ is given by a set of smooth
global vector fields $X_{1},\dots,X_{m}$ called the generating family. The
distribution at a point $p\in M$ is
$\mathcal{D}_{p}:=\mathrm{span}\\{X_{1}(p),\dots,X_{m}(p)\\}$. The rank of the
sub-Riemannian structure at $p\in M$ is
$\mathrm{rank}(p):=\mathrm{dim}(\mathcal{D}_{p})$. Observe that in our
definition, a sub-Riemannian manifold may be rank-varying. An inner product on
$\mathcal{D}_{p}$ is induced by the polarisation formula applied to the norm

$\|u\|^{2}_{\mathcal{D}_{p}}:=\min\left\\{\sum_{k=1}^{m}u^{2}_{i}\ \middle|\
\sum_{k=1}^{m}u_{i}X_{k}(p)=u\right\\}.$

A curve $\gamma:\interval{0}{T}\to M$ with initial value $\gamma(0)=p\in M$ is
horizontal if there exists
$u\in\mathrm{L}^{2}(\interval{0}{T},\mathbb{R}^{m})$, called a control, such
that $\dot{\gamma}(t)=\sum_{k=0}^{m}u_{k}(t)X_{k}(\gamma(t))$ for almost every
$t\in\interval{0}{T}$. In fact, from Carathéodory’s existence theorem, there
exists a unique maximal Lipschitz solution to the Cauchy problem

$\begin{cases}\dot{\gamma}(t)=\sum_{k=0}^{m}u_{k}(t)X_{k}(\gamma(t))\\\
\gamma(0)=p\end{cases}$ (1)

for every $u\in\mathrm{L}^{2}(\interval{0}{T},\mathbb{R}^{m})$ and $p\in M$.

The sub-Riemannian length and the sub-Riemannian energy of $\gamma$ are
defined by

$\mathrm{L}(\gamma)=\int_{0}^{T}\lVert\dot{\gamma}(t)\rVert_{\mathcal{D}_{\gamma(t)}}\mathop{}\\!\mathrm{d}t,\
\mathrm{J}(\gamma)=\dfrac{1}{2}\int_{0}^{T}\lVert\dot{\gamma}(t)\rVert^{2}_{\mathcal{D}_{\gamma(t)}}\mathop{}\\!\mathrm{d}t.$
(2)

In the case where every two points can be joined by a horizontal curve, we
have a well-defined distance function on $M$.

######  Definition 4.

The distance function of a sub-Riemannian manifold $M$, also called the
Carnot-Carathéodory distance, is defined by

$\mathop{}\\!\mathrm{d}(x,y):=\inf\\{\mathrm{L}(\gamma)\mid\gamma:[0,T]\to
M\text{ is horizontal and }\gamma(0)=x\text{ and }\gamma(T)=y\\}.$

In this work, we assume that the sub-Riemannian structures satisfy the
Hörmander condition, that is to say
$\operatorname{\mathrm{Lie}}_{p}(\mathcal{D})=\operatorname{\mathrm{T}}_{p}(M)$
for all $p\in M$, where $\operatorname{\mathrm{Lie}}_{p}(\mathcal{D})$ is
defined as the smallest Lie algebra equipped with the Lie bracket of vector
fields that contains $\mathcal{D}$. In that case, we also say that
$\mathcal{D}$ is bracket-generating. This is motivated by the well-known
result from Chow and Rashevskii, see [ABB20, Theorem 3.31.].

######  Theorem 5 (Chow–Rashevskii theorem).

Let $M$ be a sub-Riemannian manifold such that its distribution $\mathcal{D}$
is $\mathcal{C}^{\infty}$ and satisfies the Hörmander condition. Then,
$(M,\mathop{}\\!\mathrm{d})$ is a metric space and the manifold and metric
topology of $M$ coincide.

On the space of controls $\mathrm{L}^{2}(\interval{0}{T},\mathbb{R}^{m})$, we
can define a length functional, as well as a corresponding energy functional

$\mathrm{L}(u):=\int_{0}^{T}\|u(t)\|_{\mathbb{R}^{m}}\mathop{}\\!\mathrm{d}t,\
\mathrm{J}(u):=\dfrac{1}{2}\int_{0}^{T}\|u(t)\|^{2}_{\mathbb{R}^{m}}\mathop{}\\!\mathrm{d}t.$

Given an horizontal curve $\gamma:\interval{0}{T}\to M$, we define at every
differentiability point of $\gamma$ the minimal control $\overline{u}$
associated with $\gamma$

$\overline{u}(t):=\arg\min\left\\{\|u\|_{\mathbb{R}^{m}}\mid
u\in\mathbb{R}^{m},\dot{\gamma}(t)=\sum_{k=0}^{m}u_{k}X_{k}(\gamma(t))\right\\}.$

The relationship with the functionals defined in Equation 2 is the following:
$\mathrm{L}(\gamma)=\mathrm{L}(\overline{u})$ and
$\mathrm{J}(\gamma)=\mathrm{J}(\overline{u})$ when $\overline{u}$ is the
minimal control associated with $\gamma$. Through the Cauchy problem (1), it
can be seen that finding a length minimiser for $\mathrm{L}$ among the
horizontal curves with fixed end-points $\gamma(0)=p$ and $\gamma(T)=q$ is
equivalent to finding a minimal control for $L$ for which the associated path
joins $p$ and $q$. Furthermore, we have the following classical
correspondence: a horizontal curve $\gamma:\interval{0}{T}\to M$ joining $p$
to $q$ is a minimiser of $\mathrm{J}$ if and only if it is a minimiser of
$\mathrm{L}$ and is parametrised by constant speed.

Now that we can turn a sub-Riemannian manifold into a metric space, we would
like to study the geodesics associated with its distance function. These would
be horizontal curves that are locally minimising the sub-Riemannian length
functional. Because of the lack of a torsion-free metric connection, we cannot
have a geodesic equation through a covariant derivative. Rather, sub-
Riemannian geodesic are characterised via Hamilton’s equations.

The Hamiltonian of the sub-Riemannian structure is defined by

$H:\operatorname{\mathrm{T}}^{*}(M)\to\mathds{R}:\lambda\mapsto
H(\lambda):=\max_{u\in\mathbb{R}^{m}}\left(\sum_{k=1}^{m}u_{k}\langle\lambda,X_{k}(\pi(\lambda))\rangle-\dfrac{1}{2}\sum_{k=1}^{m}u_{k}^{2}\right).$
(3)

The Hamiltonian $H$ may written in terms of the generating family of the sub-
Riemannian structure $(X_{1},\dots,X_{m})$ as follows

$H(\lambda)=\frac{1}{2}\sum_{k=1}^{m}h_{k}(p,\lambda_{0})^{2},\qquad\forall\lambda\in
T^{*}(M),$

where $h_{k}(p,\lambda_{0}):=\langle\lambda_{0},X_{k}(p)\rangle$. For $p\in
M$, we will also write $H_{p}$ for the restriction of $H$ to the cotangent
space $\operatorname{\mathrm{T}}_{p}^{*}(M)$. The following result can also be
seen as an application of the Maximum Principle of Pontryagin to the sub-
Riemannian energy minimisation problem.

######  Theorem 6 (Pontryagin’s Maximum Principle).

Let $\gamma:\interval{0}{T}\to M$ be a horizontal curve which is a length
minimiser among horizontal curves, and parametrised by constant speed. Then,
there exists a Lipschitz curve $\lambda(t)\in\mathrm{T}^{*}_{\gamma(t)}(M)$
such that one and only one of the following is satisfied:

  1. (N)

$\dot{\lambda}=\overrightarrow{H}(\lambda)$, where $\overrightarrow{H}$ is the
unique vector field in $\mathrm{T}^{*}(M)$ such that
$\omega(\cdot,\overrightarrow{H}(\lambda))=\mathrm{d}_{\lambda}H$ for all
$\lambda\in\mathrm{T}^{*}(M)$ and $\omega$ denotes the canonical symplectic
form on the cotangent bundle $\operatorname{\mathrm{T}}^{*}(M)$;

  2. (A)

$\omega_{\lambda(t)}(\dot{\lambda}(t),\cap_{k=1}^{n}\mathrm{ker}(\mathrm{d}_{\lambda(t)}h_{k}))=0$
for all $t\in\interval{0}{T}$.

Moreover, in case (A), we have $\lambda(t)\neq 0$ for every
$t\in\interval{0}{T}$.

If $\lambda$ is a curve in $\operatorname{\mathrm{T}}^{*}(M)$ that satisfies
$(N)$ (resp. $(A)$), we will also say that $\lambda$ is a normal extremal
(resp. abnormal extremal). However, the projection of an abnormal extremal to
$M$ might not be locally minimising. The same terminology is used for the
corresponding curve $\gamma=\pi(\lambda)$ and for the minimal control
associated to $\gamma$. Here, the map $\pi:\mathrm{T}^{*}(M)\to M$ denotes the
bundle projection. We note that an extremal could be both normal and abnormal.
A normal trajectory $\gamma:\interval{0}{T}\to M$ is called strictly normal if
it is not abnormal. If, in addition, the restriction
$\gamma|_{\interval{0}{s}}$ is strictly normal for every $s>0$, we say that
$\gamma$ is strongly normal. It can be seen that $\gamma$ is strongly normal
if and only if the normal geodesic $\gamma$ does not contain any abnormal
segment. The projection of a normal extremal to $M$ is locally minimising,
that is to say it is a geodesic (for the sub-Riemannian distance) parametrised
by constant-speed. More precisely, it holds
$\frac{1}{2}\|\dot{\gamma}(t)\|^{2}=H(\lambda(t))$ for every
$t\in\interval{0}{T}$.

Sub-Riemannian structures which do not admit any non-trivial (i.e. non-
constant) abnormal geodesics (the trivial geodesic is always abnormal as soon
as $\mathrm{rank}(\mathcal{D}_{p})<\mathrm{dim}(M)$) are said to be _ideal_.

The theory of ordinary differential equations proves the existence of a
maximal solution to $(N)$ for every given initial condition
$\lambda(0)\in\operatorname{\mathrm{T}}^{*}(M)$. The time $t$ \- flow of
Hamilton’s equation $(N)$ is the semigroup denoted by
$\mathrm{e}^{t\overrightarrow{H}}:\operatorname{\mathrm{T}}^{*}(M)\to\operatorname{\mathrm{T}}^{*}(M)$.
The restriction of the time $1$ \- map of this flow to the fibre
$\operatorname{\mathrm{T}}^{*}_{p}(M)$, followed by projection to the base, is
called the sub-Riemannian exponential map based at $p$.

######  Definition 7.

The sub-Riemannian exponential map at $p\in M$ is the map

$\mathrm{exp}_{p}:U_{p}\to
M:\lambda\mapsto\pi(\mathrm{e}^{\overrightarrow{H}}(\lambda)),$

where $U_{p}\subseteq\operatorname{\mathrm{T}}^{*}_{p}(M)$ is the open set of
covectors such that the corresponding solution of $(N)$ is defined on the
interval $\interval{0}{1}$.

The sub-Riemannian exponential map $\mathrm{exp}_{p}$ is smooth. If
$\lambda:\interval{0}{T}\to\operatorname{\mathrm{T}}^{*}(M)$ is the normal
extremal that satisfies the initial condition
$\lambda(0)=(p,\lambda_{0})\in\operatorname{\mathrm{T}}^{*}(M)$, then the
corresponding normal extremal path $\gamma(t)=\pi(\lambda(t))$ by definition
satisfies $\gamma(t)=\mathrm{exp}_{p}(t\lambda_{0})$ for all
$t\in\interval{0}{T}$. If $M$ is complete for the Carathéodory distance, then
$U_{p}=\operatorname{\mathrm{T}}^{*}_{p}(M)$, and if in addition there are no
stricly abnormal length minimisers, then the exponential map
$\mathrm{exp}_{p}$ is surjective. Contrary to the Riemannian case, the sub-
Riemannian exponential map is not necessarily a diffeomorphism of a small ball
in $\operatorname{\mathrm{T}}^{*}_{p}(M)$ onto a small geodesic ball in $M$.
In fact,
$\operatorname{\mathrm{Im}}(\mathop{}\\!\mathrm{d}_{0}\mathrm{exp}_{p})=\mathcal{D}_{p}$
and $\mathrm{exp}_{p}$ is a local diffeomorphism at
$0\in\operatorname{\mathrm{T}}^{*}_{p}(M)$ if and only if
$\mathcal{D}_{p}=\operatorname{\mathrm{T}}_{p}(M)$.

######  Definition 8.

The critical points of the exponential map $\mathrm{exp}_{p}:U_{p}\to M$ are
called conjugate covectors at $p$. We denote by $\mathrm{Conj}(p)\subseteq
U_{p}$ the collection of all such covectors. If
$s\lambda_{0}\in\mathrm{T}^{*}_{p}(M)$ is conjugate, we say that the point
$q:=\mathrm{exp}_{p}(s\lambda_{0})$ is conjugate to
$p=\mathrm{exp}_{p}(0\cdot\lambda_{0})$ along the normal geodesic
$\gamma(t):=\mathrm{exp}_{p}(t\lambda_{0}),$ and that $s$ is a conjugate time.

We mention here some important properties related to conjugacy. The
restriction of a normal extremal $\gamma$ to an interval
$\interval{t}{t+\varepsilon}$ is abnormal if and only if $\gamma(s)$ is a
conjugate point to $\gamma(0)$ for all $s\in\interval{t}{t+\varepsilon}$
([ABB20, Theorem 8.47]). Furthermore, the set of conjugate times of $\gamma$
is discrete if $\gamma$ does not contain does not contain abnormal segments
([ABB20, Corollary 8.51]). We will also use the following result from [ABB20,
Theorem 8.61.].

######  Theorem 9.

Let $\gamma:\interval{0}{T}\to M$ be a normal extremal that does not contain
abnormal segments. If $\gamma$ has no conjugate points, then it is a local
minimiser for the length on the space of admissible trajectories with the same
endpoints. If $\gamma$ contains at least one conjugate point to $\gamma(0)$,
then it is not a local minimiser on the space of admissible trajectories with
the same endpoints.

##  3 Non-local injectivity of the sub-Riemannian exponential map

###  3.1 A family of normal extremals and the associated flow

For the rest of this work, we fix a sub-Riemannian manifold $M$ and a point
$p\in M$.

######  Definition 10.

The family of extremals from $p$ in augmented space is the map given by

$F_{p}:\mathcal{U}_{p}\to\mathds{R}\times
M:(t,\lambda_{0})\mapsto(t,\mathrm{exp}_{p}(t\lambda_{0})),$

where $\mathcal{U}_{p}\subseteq\mathds{R}\times\mathrm{T}^{*}_{p}(M)$ is the
maximal open set on which $F_{p}$ is well-defined.

The non-local injectivity of the exponential map may be related to the non-
local injectivity of the family of extremals.

######  Definition 11.

We say that the family of extremals containing $p\in M$ simply covers
$(t,q)\in\mathds{R}\times M$ if there exists an open
$\mathcal{U}^{\prime}\subseteq\mathcal{U}_{p}$ such that $F_{p}$ is injective
on $\mathcal{U^{\prime}}$ and $(t,q)\in
F_{p}(\mathcal{U^{\prime}})\subseteq\mathds{R}\times M$.

######  Proposition 12.

The exponential map $\mathrm{exp}_{p}:U_{p}\to M$ is injective in a
neighbourhood of a covector $\lambda_{0}\in
U_{p}\subseteq\mathrm{T}^{*}_{p}(M)$ if and only if the family of extremals
$F_{p}$ simply covers $(1,\mathrm{exp}_{p}(\lambda_{0}))\in\mathds{R}\times
M$.

###### Proof.

Suppose that $\mathrm{exp}_{p}$ is injective in a neighbourhood
$\lambda_{0}\in U^{\prime}\subseteq U_{p}$. Then for small $\varepsilon>0$,
the restriction of $F_{p}$ to $\ointerval{1-\varepsilon}{1+\varepsilon}\times
U^{\prime}$ is also injective. Indeed, the equality
$F_{p}(t,\lambda_{0})=F_{p}(t^{\prime},\lambda_{0}^{\prime})$, for some
$t,t^{\prime}\in\ointerval{1-\varepsilon}{1+\varepsilon}$ and
$\lambda_{0},\lambda_{0}^{\prime}\in U^{\prime}$, implies that $t=t^{\prime}$,
$\mathrm{exp}_{p}(t\lambda_{0})=\mathrm{exp}_{p}(t^{\prime}\lambda_{0}^{\prime})$,
and thus $\lambda_{0}=\lambda_{0}^{\prime}$. The other implication is proved
in a similar fashion. ∎

Similarly, we may parametrise the lift of normal extremals.

######  Definition 13.

The flow of extremals based at $p\in M$ is the map given by

$\Phi_{p}:\mathcal{V}_{p}\to\mathds{R}\times\mathrm{T}^{*}(M):(t,\lambda_{0})\mapsto(t,\mathrm{e}^{t\overrightarrow{H}}(p,\lambda_{0})),$

where $\mathcal{V}_{p}\subseteq\mathds{R}\times\mathrm{T}^{*}_{p}(M)$ is the
maximal open set on which $\Phi_{p}$ is well-defined.

######  Remark 14.

From the definition of $\mathrm{exp}_{p}(\lambda_{0})$ and
$\mathrm{e}^{t\overrightarrow{H}}(p,\lambda_{0})$, we know that the domains of
the maps $F_{p}$ and $\Phi_{p}$ coincide: $\mathcal{U}_{p}=\mathcal{V}_{p}$.

The maps $F_{p}$ and $\Phi_{p}$ defined in this section are analogues of the
concept of a family of extremals in the classical calculus of variations.
Using the standard terminology, the family of extremals considered here is
central since the extremals all start from the same point $p$.

A cotangent version of Gauss’ lemma reads in this context as follows, see
[ABB20, Proposition 8.42].

######  Theorem 15.

Let $p\in M$, $t\in\mathds{R}$, and
$\delta:\ointerval{-\varepsilon}{\varepsilon}\to\mathrm{T}^{*}_{p}(M)$ such
that $t\delta(s)\in U_{p}$ for all
$s\in\ointerval{-\varepsilon}{\varepsilon}$. Let
$\lambda(t,s):=\mathrm{e}^{t\overrightarrow{H}}(p,\delta(s))$ and
$\gamma(t,s):=\pi(\lambda(t,s))=\mathrm{exp}_{p}(t\delta(s))$,

$\left\langle\lambda(t,s),\frac{\mathrm{d}}{\mathrm{d}s}\gamma(t,s)\right\rangle=\frac{\mathrm{d}}{\mathrm{d}s}H(\lambda(t,s)),$

where $\langle\cdot,\cdot\rangle$ denotes the natural pairing of covectors and
vectors.

###  3.2 Hilbert invariant integrals

The Poincaré-Cartan one-form on $\mathds{R}\times\mathrm{T}^{*}(M)$,
associated to the Hamiltonian of the sub-Riemannian structure, is defined to
be

$\theta-H\mathrm{d}t\in\Omega^{1}(\mathds{R}\times\mathrm{T}^{*}(M)),$

where $\theta\in\Omega^{1}(\mathrm{T}^{*}(M))$ denotes the tautological one-
form of the cotangent bundle. The pullback map associated to the smooth map
$\Phi_{p}$ will be denoted by $\Phi^{*}_{p}$.

######  Definition 16.

The Hilbert integral $I^{*}_{p}$, defined on an open set
$\mathcal{U}\subseteq\mathcal{U}_{p}\subseteq\mathds{R}\times\operatorname{\mathrm{T}}^{*}_{p}(M)$,
is the line integral obtained by integrating the one-form
$\eta^{*}_{p}:=\Phi_{p}^{*}(\theta-H\mathrm{d}t)\in\Omega^{1}(\mathcal{U})$.
In other words, if $\Gamma^{*}$ is a smooth parametrised curve in
$\mathcal{U}$, then

$I^{*}_{p}[\Gamma^{*}]=\int_{\Gamma^{*}}\eta^{*}_{p}=\int_{\Gamma^{*}}\Phi_{p}^{*}(\theta-H\mathrm{d}t).$

Let us give precise description to the action of the form $\eta_{p}^{*}$. Fix
$(t,\lambda_{0})\in\mathcal{U}$ and
$v\in\mathrm{T}_{(t,\lambda_{0})}(\mathcal{U})$, which we identify with
$\mathrm{T}_{(t,\lambda_{0})}(\mathds{R}\times\mathrm{T}^{*}_{p}(M))$. As
usual, we set $\lambda(t):=\mathrm{e}^{t\overrightarrow{H}}(p,\lambda_{0})$
and $\gamma(t):=\mathrm{exp}_{p}(t\lambda_{0})$. By definition, we have that

$(\eta^{*}_{p})_{(t,\lambda_{0})}[v]=(\theta-H\mathrm{d}t)_{(t,\lambda(t))}\left[\mathrm{d}_{(t,\lambda_{0})}\Phi_{p}(v)\right].$

If we write $v=w+s\tfrac{\partial}{\partial t}{(t,\lambda_{0})}$ for unique
$s\in\mathds{R}$ and $w\in\mathrm{T}_{\lambda_{0}}(\mathrm{T}^{*}_{p}(M))$,
seen as a subspace of
$\mathrm{T}_{(t,\lambda_{0})}(\mathds{R}\times\mathrm{T}^{*}_{p}(M))$, we can
then express
$\mathrm{d}_{(t,\lambda_{0})}\Phi_{p}(v)\in\mathrm{T}_{(t,\lambda(t))}(\mathds{R}\times\mathrm{T}^{*}(M))$
as

$\mathrm{d}_{(t,\lambda_{0})}\Phi_{p}(v)=\mathrm{d}_{(p,\lambda_{0})}\mathrm{e}^{t\overrightarrow{H}}\circ\mathrm{d}_{\lambda_{0}}\iota_{p}(w)+s\left(\frac{\partial}{\partial
t}(t,\lambda(t))+\overrightarrow{H}(\lambda(t))\right),$ (4)

where $\iota_{p}:\mathrm{T}^{*}_{p}(M)\to\mathrm{T}^{*}(M)$ is the injection
$\iota_{p}(\lambda_{0}):=(p,\lambda_{0})$, satisfying $\pi\circ\iota_{p}=p$
for all $p\in M$. Now by recalling Hamilton’s equation from Item (N), we
obtain

$\displaystyle(\eta^{*}_{p})_{(t,\lambda_{0})}[v]$
$\displaystyle=\left\langle\lambda(t),\mathrm{d}_{\lambda_{0}}\mathrm{exp}_{p}(t\
\cdot)[w]\right\rangle+s\big{(}\langle\lambda(t),\dot{\gamma}(t)\rangle-H(\lambda(t))\big{)}.$
(5)

The Hilbert integral $I^{*}_{p}$ has some useful properties. When evaluated
along a ray in the augmented space $\mathds{R}\times\mathrm{T}^{*}_{p}(M)$, it
evaluates the length of the corresponding extremal in $M$.

######  Proposition 17.

Given an open neighbourhood $\mathcal{U}$ in $\mathcal{U}_{p}$, a covector
$\lambda_{0}\in\mathrm{T}^{*}_{p}(M)$, and the curve
$\Gamma^{*}:\interval{t_{0}}{t_{1}}\to\mathds{R}\times\operatorname{\mathrm{T}}_{p}^{*}(M)$
defined by $\Gamma^{*}(t)=(t,\lambda_{0})$, then we have

$I^{*}_{p}[\Gamma^{*}]=\mathrm{L}(\gamma|_{\interval{t_{0}}{t_{1}}}),$

where $\gamma(t):=\mathrm{exp}_{p}(t\lambda_{0})$, whenever
$\Gamma^{*}(t)\in\mathcal{U}$ and $t\lambda_{0}\in U_{p}$ for all
$t\in\interval{t_{0}}{t_{1}}$.

###### Proof.

Since $\dot{\Gamma}^{*}(t)=(1,0)$, Equation 5 implies that

$I^{*}_{p}[\Gamma^{*}]=\int_{t_{0}}^{t_{1}}\langle\lambda(t),\dot{\gamma}(t)\rangle-H(\lambda(t))\mathrm{d}t,$

where $\lambda(t):=\mathrm{e}^{t\overrightarrow{H}}(p,\lambda_{0})$. By
Equation 3, we can write the maximised Hamiltonian as

$H(\lambda(t))=\langle\lambda(t),\dot{\gamma}(t)\rangle-\frac{1}{2}|u(t)|^{2},\text{
for all }t\in\interval{t_{0}}{t_{1}},$

where $u(t)$ is the minimal control of
$\gamma(t)=\mathrm{exp}_{p}(t\lambda_{0})$, i.e.
$u_{i}(t)=\langle\lambda(t),X_{i}(\gamma(t))\rangle$ for $i=1,\dots,N$.
Therefore,

$I^{*}_{p}[\Gamma^{*}]=\int_{t_{0}}^{t_{1}}\left[\langle\lambda(t),\dot{\gamma}(t)\rangle-\left(\langle\lambda(t),\dot{\gamma}(t)\rangle-\frac{1}{2}|u(t)|^{2}\right)\right]\mathrm{d}t=\frac{1}{2}\int_{t_{0}}^{t_{1}}|u(t)|^{2}\mathrm{d}t=\mathrm{L}(\gamma|_{\interval{t_{0}}{t_{1}}}).$

∎

Our goal now is to establish that the Hilbert integral $I^{*}_{p}$ defines an
invariant integral, namely that is it is independent of path relative to
endpoints.

######  Proposition 18.

The one-form $\eta^{*}_{p}$ defined on an open subset $\mathcal{U}$ of
$\mathcal{U}_{p}\subseteq\mathds{R}\times\operatorname{\mathrm{T}}^{*}_{p}(M)$
is closed. Equivalently, the Hilbert integral $I^{*}_{p}$ is homotopy-
invariant in $\mathcal{U}$.

###### Proof.

We prove that $\eta_{p}^{*}$ is closed by showing that that
$\mathrm{d}\eta_{p}^{*}=0$ vanishes identically. We have

$\displaystyle\mathrm{d}\eta^{*}_{p}$
$\displaystyle=\mathrm{d}\left[\Phi_{p}^{*}(\theta-H\mathrm{d}t)\right]=\Phi_{p}^{*}\left[\mathrm{d}(\theta-H\mathrm{d}t)\right]=\Phi_{p}^{*}\left[\mathrm{d}(\theta-H\mathrm{d}t)\right]$
$\displaystyle=\Phi_{p}^{*}\left[\mathrm{d}\theta-\mathrm{d}H\wedge\mathrm{d}t\right]=\Phi_{p}^{*}\left[\omega-\mathrm{d}H\wedge\mathrm{d}t\right],$

where $\omega$ denotes the Poincaré two-form on $\mathrm{T}^{*}(M)$. If
$(t,\lambda_{0})\in\mathcal{U}\subseteq\mathds{R}\times\operatorname{\mathrm{T}}^{*}_{p}(M)$
and
$v_{1},v_{2}\in\mathrm{T}_{(t,\lambda_{0})}(\mathcal{U})\simeq\mathds{R}\times\operatorname{\mathrm{T}}^{*}_{p}(M)$,
then

$\displaystyle(\mathrm{d}\eta^{*}_{p})_{(t,\lambda_{0})}(v_{1},v_{2})$
$\displaystyle=\left[\omega-\mathrm{d}H\wedge\mathrm{d}t\right]_{(t,\lambda(t))}(\mathrm{d}_{(t,\lambda_{0})}\Phi_{p}(v_{1}),\mathrm{d}_{(t,\lambda_{0})}\Phi_{p}(v_{2})).$

As previously in Equation 4, we let $v_{i}=w_{i}+s_{i}\frac{\partial}{\partial
t}{(t,\lambda_{0})}$ for unique $s_{i}\in\mathds{R}$ and
$w_{i}\in\mathrm{T}_{\lambda_{0}}(\mathrm{T}^{*}_{p}(M))$. Without loss of
generality, we only need to treat the following two cases: $s_{1}=s_{2}=0$,
and $s_{1}=0$ while $s_{2}\neq 0$.

In the first case, we use the invariance of the symplectic form under the
Hamiltonian flow, i.e. the equality
$\mathrm{e}^{t\overrightarrow{H}*}\omega=\omega$, as well as the fact that
$\mathrm{d}_{(p,\lambda_{0})}\pi\circ\mathrm{d}_{\lambda_{0}}\iota_{p}=0$ to
find

$\displaystyle(\mathrm{d}\eta^{*}_{p})_{(t,\lambda_{0})}(v_{1},v_{2})$
$\displaystyle=\omega_{\lambda(t)}(\mathrm{d}_{(p,\lambda_{0})}\mathrm{e}^{t\overrightarrow{H}}\circ\mathrm{d}_{\lambda_{0}}\iota_{p}(w_{1}),\mathrm{d}_{(p,\lambda_{0})}\mathrm{e}^{t\overrightarrow{H}}\circ\mathrm{d}_{\lambda_{0}}\iota_{p}(w_{2}))$
$\displaystyle=(\mathrm{e}^{t\overrightarrow{H}*}\omega)_{(p,\lambda_{0})}(\mathrm{d}_{\lambda_{0}}\iota_{p}(w_{1}),\mathrm{d}_{\lambda_{0}}\iota_{p}(w_{2}))=\omega_{(p,\lambda_{0})}(\mathrm{d}_{\lambda_{0}}\iota_{p}(w_{1}),\mathrm{d}_{\lambda_{0}}\iota_{p}(w_{2}))$
$\displaystyle=(\iota_{p}\omega)^{*}_{\lambda_{0}}(w_{1},w_{2})=\iota_{p}^{*}(\mathrm{d}\theta)_{\lambda_{0}}(w_{1},w_{2})=\mathrm{d}\iota_{p}^{*}(\theta)_{\lambda_{0}}(w_{1},w_{2})=0.$

In the second case, we use Hamilton’s equation alongside the definition of the
symplectic gradient, and we obtain

$\displaystyle(\mathrm{d}\eta^{*}_{p})_{(\lambda_{0},t)}(v_{1},v_{2})$
$\displaystyle=\omega_{\lambda(t)}\left(\mathrm{d}_{(p,\lambda_{0})}\mathrm{e}^{t\overrightarrow{H}}\circ\mathrm{d}_{\lambda_{0}}\iota_{p}(w_{1}),\overrightarrow{H}(\lambda(t))\right)$
$\displaystyle\qquad-(\mathrm{d}H\wedge\mathrm{d}t)_{(t,\lambda(t))}\left(\mathrm{d}_{(p,\lambda_{0})}\mathrm{e}^{t\overrightarrow{H}}\circ\mathrm{d}_{\lambda_{0}}\iota_{p}(w_{1}),\frac{\partial}{\partial
t}(t,\lambda(t))\right)$
$\displaystyle\qquad-(\mathrm{d}H\wedge\mathrm{d}t)_{(t,\lambda(t),t)}\left(\mathrm{d}_{(p,\lambda_{0})}\mathrm{e}^{t\overrightarrow{H}}\circ\mathrm{d}_{\lambda_{0}}\iota_{p}(w_{1}),\overrightarrow{H}(\lambda(t))\right)$
$\displaystyle=(\mathrm{e}^{t\overrightarrow{H}}(p,\cdot)^{*}\mathrm{d}H)_{\lambda_{0}}(w_{1})-(\mathrm{e}^{t\overrightarrow{H}}(p,\cdot)^{*}\mathrm{d}H)_{\lambda_{0}}(w_{1})\wedge\mathrm{d}t(\overrightarrow{H}(\lambda(t)))$
$\displaystyle\qquad+(\mathrm{d}H)_{\lambda(t)}(\overrightarrow{H}(\lambda(t)))\wedge\mathrm{d}t(\mathrm{d}_{(p,\lambda_{0})}\mathrm{e}^{t\overrightarrow{H}}\circ\mathrm{d}_{\lambda_{0}}\iota_{p}(w_{1}))$
$\displaystyle\qquad-(\mathrm{e}^{t\overrightarrow{H}}(p,\cdot)^{*}\mathrm{d}H)_{\lambda_{0}}(w_{1})\wedge\mathrm{d}t\left(\frac{\partial}{\partial
t}(t,\lambda(t))\right)$
$\displaystyle\qquad+(\mathrm{d}H)_{\lambda(t)}\left(\frac{\partial}{\partial
t}(t,\lambda(t))\right)\wedge\mathrm{d}t(\mathrm{d}_{(p,\lambda_{0})}\mathrm{e}^{t\overrightarrow{H}}\circ\mathrm{d}_{\lambda_{0}}\iota_{p}(w_{1}))=0,$

which concludes the proof. ∎

We will now complete the proof of the invariance of $I^{*}_{p}$.

######  Proposition 19.

Let $\mathcal{U}\subseteq\mathds{R}\times\operatorname{\mathrm{T}}^{*}_{p}(M)$
be an open neighbourhood which is convex in the $\mathds{R}$-direction (i.e.
if $(t_{0},\lambda_{0}),(t_{1},\lambda_{0})\in\mathcal{U}$, then
$(t,\lambda_{0})\in\mathcal{U}$ for all $t\in\interval{t_{0}}{t_{1}}$). Then,
one-form $\eta^{*}_{p}$ defined on $\mathcal{U}$ is exact. Equivalently, the
Hilbert integral $I^{*}_{p}$ is path-independent in $\mathcal{U}$.

###### Proof.

Let $\Gamma^{*}(s)=(t(s),\lambda_{0}(s))$ be closed path in $\mathcal{U}$
parametrised on the interval $\interval{a}{b}$. According to the convexity
hypothesis, this curve is homotopic to the curve
$\Gamma_{0}^{*}(s):=(t(a),\lambda_{0}(s))$. It follows from Proposition 18,
Equation 5 and the cotangent version Gauss’ lemma as seen in Theorem 15 that

$\displaystyle\int_{\Gamma^{*}}\eta_{p}^{*}$
$\displaystyle=\int_{\Gamma_{0}^{*}}\eta_{p}^{*}=\int_{a}^{b}\frac{\mathrm{d}}{\mathrm{d}s}\left[H\left(\mathrm{e}^{t(a)\overrightarrow{H}}(p,\lambda_{0}(s))\right)\right]\mathrm{d}s$
$\displaystyle=H\left(\mathrm{e}^{t(a)\overrightarrow{H}}(p,\lambda_{0}(b))\right)-H\left(\mathrm{e}^{t(a)\overrightarrow{H}}(p,\lambda_{0}(a))\right)=0,$

since $\lambda_{0}(\cdot)$ is a closed curve. ∎

When a family of extremals containing $p$ simply covers an open neighbourhood
of $\mathds{R}\times M$, it can be used to define a Hilbert integral $I_{p}$
on a subset of $\mathds{R}\times M$.

######  Definition 20.

Assume that the family of extremals $F_{p}$ is injective on a neighbourhood
$\lambda_{0}\in\mathcal{U}\subseteq\mathds{R}\times\operatorname{\mathrm{T}}^{*}_{p}(M)$.
Then the Hilbert integral $I_{p}$, defined on
$F_{p}(\mathcal{U})\subseteq\mathds{R}\times M$, is the line integral obtained
by integrating the continuous one-form

$(\eta_{p})_{(t,q)}(s,w)=\langle\lambda_{t},w\rangle-H(q,\lambda_{t})s,$ (6)

where $(q,\lambda_{t}):=\mathrm{e}^{t\overrightarrow{H}}(p,\lambda_{0})$,
$(t,\lambda_{0}):=F_{p}^{-1}(t,q)$, and $s\frac{\partial}{\partial t}+w\in
T_{(t,q)}(\mathds{R}\times M).$

######  Remark 21.

While $\eta^{*}_{p}\in\Omega^{1}(\mathcal{U})$ is smooth, the form
$\eta_{p}\in\Omega^{1}(F_{p}(\mathcal{U}))$ needs to be only continuous.
Namely the functions $(q,\lambda_{t})$ in its definition depend smoothly on
$t,p,\lambda_{0}$, but the value $\lambda_{0}$ in turn depends only
continuously on $(t,q).$ This is since $F_{p}$ is only simply covering,
thereby a homeomorphism. This lack of smoothness will require a detailed
analysis, as given in Section 3.4.

In a similar way to Proposition 17, evaluating the Hilbert integral $I_{p}$ on
the graph of one of the normal extremals of the family $F_{p}$ simply gives
its length.

######  Proposition 22.

Given an open neighbourhood $\mathcal{U}\subseteq\mathcal{U}_{p}$ such that
the family of extremals $F_{p}$ is injective on $\mathcal{U}$, a covector
$\lambda_{0}\in\mathrm{T}^{*}_{p}(M)$, and the curve
$\Gamma:\interval{t_{0}}{t_{1}}\to F_{p}(\mathcal{U}_{p})$ defined by
$\Gamma(t):=(t,\mathrm{exp}_{p}(t\lambda_{0}))$. Then the following equality
holds :

$I_{p}[\Gamma]=\mathrm{L}(\gamma|_{\interval{t_{0}}{t_{1}}}).$

Here $\gamma(t):=\mathrm{exp}_{p}(t\lambda_{0})$, whenever $\Gamma(t)\in
F_{p}(\mathcal{U})$ and $t\lambda_{0}\in U_{p}$ for all
$t\in\interval{t_{0}}{t_{1}}$.

###### Proof.

Reasoning as in the proof of Proposition 17 and using Equation 6, it is seen
that

$I_{p}[\Gamma]=\int_{\Gamma}\eta_{p}=\int_{t_{0}}^{t_{1}}\left(\langle\mathrm{e}^{t\overrightarrow{H}}(p,\lambda_{0}),\dot{\gamma}(t)\rangle-H(\mathrm{e}^{t\overrightarrow{H}}(p,\lambda_{0}))\right)\mathrm{d}t=\mathrm{L}(\gamma|_{\interval{t_{0}}{t_{1}}}).$

∎

Furthermore, the Hilbert integrals and $I^{*}_{p}$ on $\mathcal{U}$ and
$I_{p}$ on $F_{p}(\mathcal{U})$ coincide on smooth curves (assuming that
$F_{p}$ is injective), as shown in the following proposition.

######  Proposition 23.

Suppose that the family of extremals $F_{p}:\mathcal{U}\to\mathds{R}\times M$
is an injective map. Assume also that either $\Gamma^{*}$ is a smooth curve in
$\mathcal{U}$ and $\Gamma:=F_{p}\circ\Gamma^{*}$ is the corresponding smooth
curve in $F_{p}(\mathcal{U})$, or assume that both $\Gamma$ is a smooth curve
in $F_{p}(\mathcal{U})$ and the corresponding curve
$\Gamma^{*}=F_{p}^{-1}\circ\Gamma$ in $\mathcal{U}$ are smooth. Then
$I^{*}_{p}[\Gamma^{*}]=I_{p}[\Gamma]$.

###### Proof.

Let us write $\Gamma(s):=(t(s),q(s))$, as well as
$\Gamma^{*}(s):=(t(s),\lambda_{0}(s))$, with
$q(s):=\mathrm{exp}_{p}(t_{0}(s)\lambda_{0}(s))$, where the functions
$t(\cdot),q(\cdot)$ and $\lambda_{0}(\cdot)$ are smooth. As before we use the
notation $\lambda(s):=\mathrm{e}^{t(s)\overrightarrow{H}}(p,\lambda_{0}(s))$
and $\gamma(s):=\mathrm{exp}_{p}(t_{0}(s),\lambda_{0}(s))$.

Notice first that

$\displaystyle\dot{\gamma}(s)$
$\displaystyle=\frac{\mathrm{d}}{\mathrm{d}s}\left(\pi\circ\mathrm{e}^{t(s)\overrightarrow{H}}(p,\lambda_{0}(s))\right)=\mathrm{d}_{\lambda(s)}\pi\left[\dot{t}(s)\overrightarrow{H}(\lambda(s))+\mathrm{d}_{\lambda_{0}(s)}\mathrm{e}^{t(s)\overrightarrow{H}}(p,\cdot)(\dot{\lambda}_{0}(s))\right].$

Therefore, Equation 5 and Equation 6 yield

$\displaystyle(\eta_{p})_{\Gamma(s)}(\dot{\Gamma}(s))$
$\displaystyle=\langle\lambda(s),\dot{\gamma}(s)\rangle-\dot{t}(s)H(\lambda(s))$
$\displaystyle=\langle\lambda(s),\mathrm{d}_{\lambda(s)}\pi\circ\mathrm{d}_{\lambda_{0}(s)}\mathrm{e}^{t(s)\overrightarrow{H}}(p,\cdot)(\dot{\lambda}_{0}(s))\rangle$
$\displaystyle\qquad+\dot{t}(s)\left[\langle\lambda(s),\mathrm{d}_{\lambda(s)}\pi\circ\overrightarrow{H}(\lambda(s))\rangle-H(\lambda(s))\right]$
$\displaystyle=(\eta^{*}_{p})_{\Gamma^{*}(t)}(\dot{\Gamma}^{*}(t)),$

which proves the statement. ∎

The relations between $I_{p}$ and $I^{*}_{p}$ given by the last two
propositions are insufficient to extract the exactness of $\eta_{p}$. In order
to progress, we will now study the nature of the sub-Riemannian conjugate
locus.

###  3.3 The sub-Riemannian conjugate locus

Let us recall the regularity and continuity properties of the sub-Riemannian
exponential map proven by the authors in [BK21]. If a Riemannian metric $g$ on
$M$ is fixed, one can consider the isomorphism

$\sharp:\mathrm{Ver}_{\lambda}\subseteq\operatorname{\mathrm{T}}_{\lambda}(\operatorname{\mathrm{T}}^{*}_{p}(M))\to\mathrm{T}_{p}(M):\xi\mapsto\xi^{\sharp},\qquad
p:=\pi(\lambda),\ \lambda\in\mathrm{T}^{*}(M),$

where $\xi^{\sharp}$ is the unique element of $\mathrm{T}_{p}(M)$ such that
$g(\xi^{\sharp},X)=\xi(X)$, for every $X\in\mathrm{T}_{p}(M)$, the spaces
$\mathrm{Ver}_{\lambda}$ and $\mathrm{T}^{*}_{p}(M)$ being canonically
identified. As explained in [BK21, Section 3.2], fixing a Riemannian metric in
this way is equivalent to choosing a symplectic moving frame along a normal
geodesic $\gamma(t)$, and extend the scalar product
$\langle\cdot,\cdot\rangle_{\gamma(t)}$ along $\gamma(t)$ to the whole
manifold.

The ray in $U_{p}\subseteq\operatorname{\mathrm{T}}_{p}^{*}(M)$ through
$\lambda_{0}$ is the map

$\mathrm{r}_{p,\lambda_{0}}:I_{p,\lambda_{0}}\to\mathrm{T}^{*}_{p}(M):t\mapsto
t\lambda_{0}$

where $I_{p,\lambda_{0}}\subseteq\mathbb{R}^{+}$ is the maximal interval
containing 0 such that $t\lambda_{0}\in U_{p}$ for every $t\in
I_{t,\lambda_{0}}$. In this way,
$\mathrm{\dot{r}}_{p,\lambda_{0}}(t)\in\operatorname{\mathrm{T}}_{t\lambda_{0}}(\operatorname{\mathrm{T}}^{*}_{p}(M))$
and identifying
$\operatorname{\mathrm{T}}_{t\lambda_{0}}(\operatorname{\mathrm{T}}^{*}_{p}(M))$
with $\operatorname{\mathrm{T}}^{*}_{p}(M)$ in the usual way, we have
$\mathrm{\dot{r}}_{p,\lambda_{0}}(t)=\lambda_{0}$ for every $t\in
I_{p,\lambda_{0}}$.

######  Theorem 24 (Warner-regularity of the sub-Riemannian exponential map
[BK21]).

Let $M$ be a sub-Riemannian manifold and $p\in M$. Then,

  1. (R1)

The map $\mathrm{exp}_{p}$ is $\mathcal{C}^{\infty}$ on
$U_{p}=\mathrm{Dom}(\mathrm{exp}_{p})$ and, for all $\lambda_{0}\in
U_{p}\setminus H^{-1}_{p}(0)$ and all $t\in I_{p,\lambda_{0}}$, we have
$\mathrm{d}_{t\lambda_{0}}\mathrm{exp}_{p}(\mathrm{\dot{r}}_{p,\lambda_{0}}(t))\neq
0_{\mathrm{exp}_{p}(t\lambda_{0})}$;

  2. (R2)

The map

$\mathrm{Ker}(\mathrm{d}_{\lambda_{0}}\mathrm{exp}_{p}(\lambda_{0}))\to\mathrm{T}_{\mathrm{exp}_{p}(\lambda_{0})}(M):\xi_{0}\mapsto\left(\mathrm{d}_{\lambda_{0}}\mathrm{e}^{t\overrightarrow{H}}[\xi_{0}]\right)^{\sharp}$

has its image $g$-perpendicular to
$\mathrm{Im}(\mathrm{d}_{\lambda_{0}}\mathrm{exp}_{p})$

  3. (R3)

Let $\lambda_{0}\in U_{p}\setminus H_{p}^{-1}(0)$ be a covector such that the
corresponding geodesic $\gamma(t):=\mathrm{exp}_{p}(t\lambda_{0})$ is strongly
normal. Then, there exists a radially convex neighbourhood $\mathcal{V}$ of
$\lambda_{0}$ such that for every ray $\mathrm{r}_{p,\overline{\lambda}_{0}}$
which intersects $\mathcal{V}$ that does not contain abnormal subsegments in
$\mathcal{V}$, the number of singularities of $\mathrm{exp}_{p}$ (counted with
multiplicities) on
$\mathrm{Im}(\mathrm{r}_{p,\overline{\lambda}_{0}})\cap\mathcal{V}$ is
constant and equals the order of $\lambda_{0}$ as a singularity of
$\mathrm{exp}_{p}$, i.e.
$\mathrm{dim}(\mathrm{Ker}(\mathop{}\\!\mathrm{d}_{\lambda_{0}}\mathrm{exp}_{p}))$.

In view of the continuity property (R3), we will now adapt Warner’s definition
of singular and regular conjugate points and their order (see [War65, Section
3.]) to the sub-Riemannian setting.

######  Definition 25.

We say that a conjugate covector
$\lambda_{0}\in\operatorname{\mathrm{T}}^{*}_{p}(M)$ is regular for
$\mathrm{exp}_{p}$ if there is a neighbourhood $U$ of $\lambda_{0}$ such that
every ray of $\mathrm{T}^{*}_{p}(M)$ contains at most one covector in $U$
which is conjugate. A conjugate covector that is not regular is said to be
singular. The collection of regular conjugate (resp. singular conjugate)
covectors is denoted by
$\mathrm{Conj}^{R}(p)\subseteq\operatorname{\mathrm{T}}^{*}_{p}(M)$ (resp.
$\mathrm{Conj}^{S}(p)$). We write
$\mathrm{Conj}^{R}_{U}(p):=\mathrm{Conj}^{R}(p)\cap U$ (resp.
$\mathrm{Conj}^{S}_{U}(p):=\mathrm{Conj}^{S}(p)\cap U$).

In Riemannian geometry, the Jacobi fields along a given geodesic satisfy a
second order differential equation. Consequently, the order of vanishing the
function $t\mapsto\mathrm{det}(\mathrm{d}_{tv}\mathrm{exp}_{p})$ at $t=1$ is
always finite and equal to the order of $v\in\mathrm{T}_{p}(M)$ as a
singularity of $\mathrm{exp}_{p}$ (see e.g. [Mor31]). In sub-Riemannian
geometry however, it is possible that this function and all its derivatives
vanishes when $t=1$. This motivates the following definition.

######  Definition 26.

We say that a covector $\lambda_{0}\in U_{p}$ has order
$m\in\mathds{N}\cup\left\\{+\infty\right\\}$ if the map
$t\mapsto\det(\mathrm{d}_{t\lambda_{0}}\mathrm{exp}_{p})$ vanishes of order
$m$ at $t=1$.

The definition above is well posed thanks to (R2) of Theorem 24. Indeed, the
domain the map

$\mathrm{d}_{t\lambda_{0}}\mathrm{exp}_{p}:\mathrm{T}_{t\lambda_{0}}(\mathrm{T}^{*}_{p}(M))\to\mathrm{T}_{\mathrm{exp}_{p}(t\lambda_{0})}(M)$

is first identified with the space $\mathrm{T}^{*}_{p}(M)$ as usual and then
we have isomorphisms

$\mathrm{T}^{*}_{p}(M)\cong\mathrm{Ker}(\mathrm{d}_{t\lambda_{0}}\mathrm{exp}_{p})\oplus\mathrm{Im}(\mathrm{d}_{t\lambda_{0}}\mathrm{exp}_{p})\cong\mathrm{T}_{\mathrm{exp}_{p}(t\lambda_{0})}(M).$

The last identification dependends on the choice of a moving frame in (R2).
However, the vanishing of the determinant of
$\mathrm{d}_{t\lambda_{0}}\mathrm{exp}_{p}$, viewed as a map from
$\mathrm{T}_{\mathrm{exp}_{p}(t\lambda_{0})}(M)$ to itself, as well as the
vanishing of its derivatives that we need for Definition 26, are independent
of this choice.

######  Proposition 27.

If $\lambda_{0}\in\mathrm{Conj}^{R}(p)$ has finite order, then there exists a
neighbourhood $U\subseteq\mathrm{T}^{*}_{p}(M)$ of $\lambda_{0}$ such that all
vectors in $\mathrm{Conj}^{R}_{U}(p)$ have the same order.

###### Proof.

Assume that $\lambda_{0}$ has order $m\geq 1$. By Malgrange’s Preparation
Theorem (see [GG73]) there exists in a neighbourhood
$\ointerval{1-\varepsilon}{1+\varepsilon}\times V$ of
$(1,\lambda_{0})\in\mathbb{R}\times\mathrm{T}^{*}_{p}(M)$ a factorisation of
the type

$\det(\mathrm{d}_{t\eta}\mathrm{exp}_{p})=c(t,\eta)\cdot\big{[}(t-1)^{m}+a_{m-1}(\eta)(t-1)^{m-1}+\cdots+a_{0}(\eta)\big{]},$
(7)

where $c:\ointerval{1-\varepsilon}{1+\varepsilon}\times V\to\mathds{R}$,
$a_{i}:V\to\mathds{R}$ are smooth functions such that $c(1,\lambda_{0})\neq 0$
and $a_{m-1}(\lambda_{0})=\dots=a_{0}(\lambda_{0})=0$. Since $\lambda_{0}$ is
regular, then for sufficiently small $\varepsilon^{\prime}\leq\varepsilon$ and
$V^{\prime}\subseteq V$, we may assume that all the conjugate covectors in
$V^{\prime}$ are regular.

If $\lambda^{\prime}_{0}\in\mathrm{Conj}(p)$ has order $m^{\prime}\geq 1$ such
that $(1,\lambda^{\prime}_{0})\in\mathcal{V}^{\prime}$, we can use Malgrange’s
Preparation Theorem again at $\lambda_{0}^{\prime}$ to find a neighbouhoord
$\ointerval{1-\varepsilon^{\prime}}{1+\varepsilon^{\prime}}\times
V^{\prime}\subseteq\ointerval{1-\varepsilon}{1+\varepsilon}\times\mathcal{U}$
and smooth functions
$c^{\prime}:\ointerval{1-\varepsilon^{\prime}}{1+\varepsilon^{\prime}}\times
V^{\prime}\to\mathds{R},a_{0}^{\prime},\dots,a_{m^{\prime}-1}^{\prime}:V^{\prime}\to\mathds{R}$
such that

$\det(\mathrm{d}_{t\eta}\mathrm{exp}_{p})=c^{\prime}(t,\eta)\cdot((t-1)^{m^{\prime}}+a^{\prime}_{m^{\prime}-1}(\eta)(t-1)^{m^{\prime}-1}+\cdots+a^{\prime}_{0}(\eta)),$
(8)

for every
$(t,\eta)\in\ointerval{1-\varepsilon^{\prime}}{1+\varepsilon^{\prime}}\times
U^{\prime}$, with $c^{\prime}(1,\lambda_{0}^{\prime})\neq 0$,
$a^{\prime}_{0}(\lambda_{0}^{\prime})=\dots=a^{\prime}_{m^{\prime}-1}(\lambda_{0}^{\prime})=0$.

Now (7) and (8) both hold in the non-empty neighbourhood
$\ointerval{1-\varepsilon^{\prime}}{1+\varepsilon^{\prime}}\times U^{\prime}$
and therefore $m$ must equal $m^{\prime}$. Indeed,
$a_{0}(\lambda_{0}^{\prime})=0$ since $\lambda_{0}^{\prime}$ is conjugate and
the limit

$\lim_{t\to
1}\frac{(t-1)^{m}+a_{m-1}(\lambda_{0}^{\prime})(t-1)^{m-1}+\cdots+a_{1}(\lambda_{0}^{\prime})(t-1)}{(t-1)^{m^{\prime}}}$

must exist and tend to
$c^{\prime}(1,\lambda_{0}^{\prime})/c(1,\lambda_{0}^{\prime})$. This implies
that $m=m^{\prime}$.

On the other hand, if a sequence $(\lambda_{k})_{k\in\mathds{N}}$ of conjugate
covectors that have infinite order converges to $\lambda_{0}$, then
$\lambda_{0}$ would also have an infinite order, since
$t\mapsto\det(\mathrm{d}_{t\eta}\mathrm{exp}_{p})$ is smooth for all $\eta\in
U_{p}$.
The above proves that the the set of points of constant order is open and
closed in $U_{p}$. ∎

######  Remark 28.

The notion of order finiteness that we introduce in the section is related to
those of ample and equiregular geodesics introduced in [ABR18]. Indeed, if
$\gamma(t):=\mathrm{exp}_{p}(t\lambda^{\prime}_{0})$ is ample and equiregular
for all $\lambda_{0}^{\prime}$ in a neighbourhood of $\lambda_{0}$ in
$\mathrm{T}^{*}_{p}(M)$, then $\lambda_{0}\in\mathrm{Conj}^{R}(p)$ must have
finite order. In general, the set of $\mathcal{A}\subseteq\mathrm{T}^{*}(M)$
of $(p,\lambda_{0})$ such that the corresponding normal geodesic is ample and
equiregular is non-empty and dense (see [ABP19, Proposition 7.1.]).

In the following theorem, we prove that in the neighbourhood of a conjugate
covector that has finite order, the conjugate locus can be given a structure
of submanifold.

######  Theorem 29.

If $\lambda_{0}\in\mathrm{T}^{*}_{p}(M)$ is a regular conjugate covector that
has finite order $m$, then there exists a neighbourhood $U$ of $\lambda_{0}$
such that $\mathrm{Conj}_{U}(p)$ is a submanifold of codimension one in
$\mathrm{T}^{*}_{p}(M)$. Moreover,
$\mathrm{T}_{\lambda_{0}}(\mathrm{T}^{*}_{p}(M))=\mathrm{T}_{\lambda_{0}}(\mathrm{Conj}_{U}(p))\oplus\mathrm{Im}(\mathrm{r}_{p,\lambda_{0}})$.

###### Proof.

Consider the smooth function

$\Delta^{m-1}_{p}:U_{p}\to\mathds{R}:\lambda\mapsto\frac{\mathrm{d}^{m-1}}{\mathrm{d}t^{m-1}}\Big{|}_{t=1}\left(\det\mathrm{d}_{t\lambda}\mathrm{exp}_{p}\right).$

From Proposition 27, we can find a neighbourhood $U\subseteq U_{p}$ of
$\lambda_{0}$ such that all covectors in $\mathrm{Conj}_{U}(p)$ are regular
and of order $m$. The inclusion
$\mathrm{Conj}_{U}(p)\subseteq(\Delta^{m-1}_{p})^{-1}(0)$ thus holds. The
derivative of $\Delta_{p}^{m-1}$ along the ray $\mathrm{r}_{p,\lambda_{0}}$ is
non-zero. Indeed, since $\lambda_{0}$ is assumed to have order $m$,

$\displaystyle\mathrm{d}_{1}\mathrm{r}_{p,\lambda_{0}}(\Delta_{p}^{m-1})$
$\displaystyle=\frac{\mathrm{d}}{\mathrm{d}s}\Big{|}_{s=1}\left(\frac{\mathrm{d}^{m-1}}{\mathrm{d}t^{m-1}}\Big{|}_{t=1}\left(\det\mathrm{d}_{ts\lambda_{0}}\mathrm{exp}_{p}\right)\right)$
$\displaystyle=(m-1)\frac{\mathrm{d}^{m-1}}{\mathrm{d}t^{m-1}}\Big{|}_{t=1}\left(\det\mathrm{d}_{t\lambda_{0}}\mathrm{exp}_{p}\right)+\frac{\mathrm{d}^{m}}{\mathrm{d}t^{m}}\Big{|}_{t=1}\left(\det\mathrm{d}_{t\lambda_{0}}\mathrm{exp}_{p}\right)$
$\displaystyle=\frac{\mathrm{d}^{m}}{\mathrm{d}t^{m}}\Big{|}_{t=1}\left(\det\mathrm{d}_{t\lambda_{0}}\mathrm{exp}_{p}\right)\neq
0.$

Therefore a possibly smaller neighborhood $U$ of $\lambda_{0}$ in
$\mathrm{T}^{*}_{p}(M)$, can be chosen to be convex and on which the radial
derivative of $\Delta_{p}^{m-1}$ is non-zero. The function $\Delta_{p}^{m-1}$
is in particular smooth on $U$ and has a non-zero differential.

Let us now show that
$(\Delta^{m-1}_{p})^{-1}(0)\subseteq\mathrm{Conj}_{U}(p)$. Suppose
$\lambda^{\prime}_{0}\in U$ and $\Delta_{p}^{m-1}(\lambda^{\prime}_{0})=0$.
The neighbourhood $U$ has the property that each ray intersecting it has
exactly one conjugate covector. Thus, on the ray passing through
$\lambda^{\prime}_{0}$, there must be a (unique) conjugate covector
$\lambda^{\prime\prime}_{0}$. Since $U$ is convex, the line joining
$\lambda^{\prime}_{0}$ to $\lambda^{\prime\prime}_{0}$ is also in $U$. The
radial derivative of $\Delta_{p}^{m-1}$ along that line is non zero while
$\Delta_{p}^{m-1}(\lambda^{\prime}_{0})=0$, by hypothesis, and
$\Delta_{p}^{m-1}(\lambda^{\prime\prime}_{0})=0$, since all the conjugate
vectors in $U$ have the same finite order. Thus, by the mean value theorem, we
must have $\lambda^{\prime}_{0}=\lambda^{\prime\prime}_{0}$, and
$\lambda^{\prime}_{0}\in\mathrm{Conj}_{U}(p)$.

Finally, by the implicit function theorem,
$(\Delta_{p}^{m-1})^{-1}(0)=\mathrm{Conj}_{U}(p)$ can be given the desired
manifold structure. ∎

######  Remark 30.

Under the hypothesis of Theorem 29, we have in particular that the set
$\Pi_{p}=\\{(t,\eta)\in\ointerval{1-\varepsilon}{1+\varepsilon}\times U\mid
t\eta\text{ is
conjugate}\\}\subseteq\mathcal{U}_{p}\subseteq\mathds{R}\times\mathrm{T}^{*}_{p}(M)$
is also a submanifold for $\varepsilon>0$ sufficiently small.

###  3.4 Non-local injectivity of the sub-Riemannian exponential map

Putting everything together, we finally prove in this section the relationship
between the injectivity of the sub-Riemannian exponential map and its
singularities.

We will need the following lemma, which is a consequence of Sard’s theorem. It
appeared in [Sav43, Lemma II], and we provide here the proof for the sake of
completeness.

######  Lemma 31.

Let $X$ and $Y$ be smooth manifolds with $\dim X=\dim Y$, and assume that $X$
is compact. If $f$ be a $C^{1}$ map from $X$ to $Y$, then almost all points of
$Y$ have a finite preimage under $f$.

###### Proof.

Let us denote by $\mathrm{Crit}(f)$ the set of critical points of $f$, i.e.

$\mathrm{Crit}(f):=\\{x\in X\mid\mathrm{d}_{x}f\text{ has a non-trivial
kernel}\\}.$

Let $y\in Y$ such that $f^{-1}(y)$ is infinite. Since $X$ is assumed to
compact, we can deduce that $f^{-1}(y)$ has at least one accumulation point
$x$. We write $(x_{n})\subseteq f^{-1}(y)$ for a sequence, distinct from $x$,
converging to $x$.

Let $\varphi$ be a coordinate chart around $x$ in $X$ and $\psi$ a coordinate
chart around $y$ in $Y$. We write $f_{\varphi\psi}=\psi\circ
f\circ\varphi^{-1}$ for the expression of the map $f$ in these coordinates.
When $n$ is large enough, we can be sure that $x_{n}\in\mathrm{Dom}(\varphi)$
and then consider the map

$f(t):\interval{0}{1}\to\mathds{R}^{\mathrm{dim}Y}:t\mapsto
f_{\varphi\psi}(\varphi(x)+t(\varphi(x_{n})-\varphi(x))).$

By the mean value theorem, there exists $t_{n}\in\ointerval{0}{1}$ such that

$f_{\varphi\psi}(\varphi(x_{n}))=f_{\varphi\psi}(\varphi(x))+\mathrm{d}_{\varphi(x)+t_{n}(\varphi(x_{n})-\varphi(x))}f_{\varphi\psi}[(\varphi(x_{n})-\varphi(x))],$

and therefore, by letting $n$ tend to $+\infty$,

$\mathrm{d}_{\varphi(x)}f_{\varphi\psi}[v]=0,\text{ where
}v=\lim_{n\to+\infty}\frac{\varphi(x_{n})-\varphi(x)}{\|\varphi(x_{n})-\varphi(x)\|}\neq
0.$

Consequently, $\mathrm{d}_{x}f$ has a non-trivial kernel and $x$ is a critical
point, i.e. $y=f(x)\in f(\mathrm{Crit}(f))$. By Sard’s theorem, the image of
$\mathrm{Crit}(f)$ under $f$ has measure zero in $Y$ and the proof is
complete. ∎

The study of the regularity of the conjugate locus in the previous section
allows us to prove that the continuous one-form $\eta_{p}$ is indeed exact.

######  Proposition 32.

If the family of extremals $F_{p}$ simply covers $(1,q)\in\mathds{R}\times M,$
where $q:=\mathrm{exp}_{p}(\lambda_{0})$ and $\lambda_{0}\in
U_{p}\subseteq\operatorname{\mathrm{T}}^{*}_{p}(M)$ is a regular conjugate
vector of finite order, then Hilbert integral $I_{p}$ is invariant on some
neighbourhood of $(1,q)$ in $\mathds{R}\times M$.

###### Proof.

By Proposition 19, the Hilbert integral $I^{*}_{p}$ is path-independent on a
small enough convex neighbourhood $\mathcal{U}$ of $(1,\lambda_{0})$ in
$\mathds{R}\times\mathrm{T}^{*}_{p}(M)$. We would like to argue that as a
consequence the same property holds for $I_{p}$.

Let $\Gamma(s)=(t(s),q(s))$ be a smooth closed curve in $F_{p}(\mathcal{U})$.
If the corresponding curve
$\Gamma^{*}(s)=(t(s),\lambda_{0}(s)):=F_{p}^{-1}(\Gamma(s))$ were to be
smooth, we would be able to conclude this by Proposition 23. However, $F_{p}$
is only a homeomorphism and thus we do not have enough regularity in general
to evaluate the line integral $I^{*}_{p}$ along $\Gamma^{*}$. This issue is
addressed by inroducing a coordinate system $\varphi$ containing $\Gamma$, and
by approximating in $C^{0}$ the loop $\varphi(\Gamma)$ by a sequence of
parametrised polygons $(p_{n})$ in $\mathds{R}^{n+1}$. These polygons may be
chosen such that they intersect the conjugate locus only a finite number of
times. Indeed, if $x\notin\Pi_{p}$, one can deduce from Lemma 31 that almost
all rays passing through $x$ in $\mathds{R}^{n+1}$ intersect $\Pi_{p}$ a
finite number of times. Finally we may conclude by Proposition 23, letting $n$
tends to $+\infty$, that

$I_{p}[\Gamma]=\lim_{n\to+\infty}I_{p}[\varphi^{-1}(p_{n})]=\lim_{n\to+\infty}I^{*}_{p}[\varphi^{-1}(p_{n})]=0,$

that is to say, $I_{p}$ is path independent. ∎

The integral $I_{p}$ is the analogy of what is called Hilbert invariant
integral in the classical calculus of variations. The fact that the family of
extremals $F_{p}$ induces the invariance of the Hilbert integral $I_{p}$
corresponds to $F_{p}$ defining what is commonly called a (central) field of
extremals. The strategy now is to deduce that subsegment of normal geodesics
are length-minimising if they are part of a field of extremals.

######  Proposition 33.

Let $\varepsilon>0$ and $\gamma:\rinterval{0}{1+\varepsilon}\to M$ be a
strongly normal extremal starting at $p\in M$ and with initial covector
$\lambda_{0}\in U_{p}$. Assume that $\lambda_{0}$ is a first conjugate
covector that is regular and has finite order. If the family of extremals
$F_{p}$ simply covers $(1,\mathrm{exp}_{p}(\lambda_{0}))$, then any subsegment
of $\gamma$ is local a length-minimiser among all the admissible trajectories
with the same endpoints.

###### Proof.

The assumption that $\lambda_{0}$ is strongly normal and is a first conjugate
covector implies, by adapting the proof of Proposition 12 with the conclusions
of Theorem 9, that the neighbourhood on which $F_{p}$ is injective can be
chosen such that it contains the ray $\mathrm{r}_{p,\lambda_{0}}$.

Consider an admissible trajectory $c:\interval{t_{0}}{t_{1}}\to M$ with
control $u^{\prime}$, different from $\gamma|_{\interval{t_{0}}{t_{1}}}$, but
with the same endpoints ($t_{0},t_{1}\in\ointerval{0}{1+\varepsilon}$). We
denote by $\Gamma^{\prime}(t)=(t,c(t))$ the corresponding curves in
$F_{p}(\mathcal{U})$. We also write $\lambda^{\prime}_{0}(t)$ for the curve of
initial covectors in $\mathrm{T}^{*}_{p}(M)$ such that
$\Gamma^{\prime}(t)=F_{p}(t,\lambda^{\prime}_{0}(t))$, as well as the lift
$\lambda^{\prime}(t):=\mathrm{e}^{t\overrightarrow{H}}(p,\lambda^{\prime}_{0}(t))$,
as usual.

By definition of the sub-Riemannian Hamiltonian, we have that

$H(\lambda^{\prime}(t))\geq\langle\lambda^{\prime}(t),\dot{c}(t)\rangle-\frac{1}{2}|u^{\prime}(t)|^{2},\text{
for almost every }t\in\interval{t_{0}}{t_{1}},$

as well as

$H(\lambda(t))=\langle\lambda(t),\dot{\gamma}(t)\rangle-\frac{1}{2}|u(t)|^{2},\text{
for all }t\in\interval{t_{0}}{t_{1}},$

since $\lambda(t)$ is the lift of the normal geodesic $\gamma$.

Therefore, we deduce that

$\frac{1}{2}\mathrm{L}(\gamma|_{\interval{t_{0}}{t_{1}}})^{2}=\frac{1}{2}\int_{t_{0}}^{t_{1}}|u(t)|^{2}\mathrm{d}t=\int_{t_{0}}^{t_{1}}\left(\langle\lambda(t),\dot{\gamma}(t)\rangle-H(\lambda(t))\right)\mathrm{d}t=I_{p}[\Gamma]=I_{p}[c]$

since the Hilbert integral $I_{p}$ is path-independent by Proposition 32, and

$\frac{1}{2}\mathrm{L}(\gamma|_{\interval{t_{0}}{t_{1}}})^{2}=\int_{t_{0}}^{t_{1}}\left(\langle\lambda^{\prime}(t),\dot{c}(t)\rangle-H(\overline{\lambda}(t))\right)\mathrm{d}t\leq\frac{1}{2}\int_{t_{0}}^{t_{1}}|u^{\prime}(t)|^{2}\mathrm{d}t=\frac{1}{2}\mathrm{L}(c)^{2}.$

∎

We can now conclude recalling from Theorem 9 that a normal extremal that does
not contain any abnormal subsegment cannot be length minimising past a
conjugate point. The proof of Theorem 1 will be completed by the following
Theorem.

######  Theorem 34.

Let $M$ be a sub-Riemannian manifold and $p\in M$. If $\lambda_{0}\in
U_{p}\subseteq\mathrm{T}^{*}_{p}(M)$ is a strongly normal and regular
conjugate covector of finite order, then the exponential map
$\mathrm{\mathrm{exp}}_{p}:U_{p}\to M$ is not injective in any neighbourhood
of $\lambda_{0}$.

###### Proof.

Without loss of generality, we may assume that $\lambda_{0}$ is a first
conjugate covector along the geodesic it generates. Since $U_{p}$ is open, the
geodesic $\gamma(t):=\mathrm{exp}_{p}(t\lambda_{0})$ is well defined for all
$t\in\rinterval{0}{1+\varepsilon}$ for some $\varepsilon>0$. If
$\mathrm{exp}_{p}$ were to be injective in some neighbourhood of
$\lambda_{0}$, then the family of extremals $F_{p}$ would simply cover
$(1,\mathrm{exp}_{p}(\lambda_{0}))$ by Proposition 12. The corresponding
Hilbert integral $I_{p}$ would be path independent by Proposition 32. Now,
Proposition 33 would imply that the normal geodesic
$\gamma:\rinterval{0}{1+\varepsilon}\to M$ is a local length minimiser among
all the admissible trajectories with the same endpoints, which would be in
contradiction with Theorem 9 since $\lambda_{0}$ is assumed to be conjugate. ∎

In the classical theory of calculus of variations (see for example [Mor96]),
the fact that an extremal is part of a field of extremals is a sufficient
condition for minimality is usually deduced from studying the Weierstrass
E-function (see [Sha66]). That argument is replaced in optimal control
problems with the maximum principle. Let us mention two instances where we
have found related uses of these techniques. In [Now88], fields of extremals
are constructed in order to formulate sufficient conditions of optimality
generalising Weierstrass’ condition. The fact that normal extremal
trajectories are locally minimising is proved by constructing a (non-central)
field of extremals in [ABB20, Section 4.7].

Let us finally discuss how the result Theorem 34 could be extended to the
whole conjugate locus. If the regular conjugate covectors of finite order is
dense in the regular conjugate locus, then by density the sub-Riemannian
exponential map will also fail to be injective in any neighbourhood of a
regular conjugate covector of infinite order. This is trivially the case if
the sub-Riemannian manifold is analytic or if any geodesic
$\gamma(t):=\mathrm{exp}_{p}(t\lambda_{0})$ is ample and equiregular for all
$(p,\lambda_{0})\in\operatorname{\mathrm{T}}^{*}(M)$. Furthermore, if the sub-
Riemannian manifold is ideal then the property (R3) from Theorem 24 implies
that regular conjugate covectors is dense in the conjugate locus, as in
[War65, Theorem 3.1].

##  4 Applications and final remarks

###  4.1 Metric geometry

The non-local injectivity of the sub-Riemannian exponential map implies a
cotangent version of Shankar-Sormani’s equivalence for synthetic notions of
conjugate points in length space (see [SS09]) along a strongly normal geodesic
with an initial covector that is regular conjugate and that has finite order.

######  Definition 35.

Let $X$ be a geodesic space and denote by $\mathcal{L}$ its length structure.
If $\gamma_{n},\gamma$ are minimising geodesics parametrised on
$\interval{0}{1}$, we say that $\gamma_{n}$ converges to $\gamma$ if they
converge for the metric

$\mathrm{d}_{\mathrm{Geo}(X)}(\gamma_{1},\gamma_{2}):=\sup_{t\in\interval{0}{1}}|\gamma_{1}(t)-\gamma_{2}(t)|+|\mathrm{L}(\gamma_{1})-\mathrm{L}(\gamma_{2})|.$

If the geodesic space in question is a sub-Riemannian manifold $M$, then it is
easy to see that a sequence of normal geodesics
$\gamma_{n}(t):=\mathrm{exp}_{p_{n}}(t\lambda_{0}^{n})$ converges to
$\gamma(t):=\mathrm{exp}_{p}(t\lambda_{0})$ if and only if $p_{n}$ converges
to $p$ in $M$ and $\lambda_{0}^{n}$ converges to $\lambda_{0}$ in
$\mathrm{T}^{*}_{p}(M)$.

######  Definition 36.

Let $X$ be a geodesic space, and a geodesic $\gamma:\interval{0}{1}\to X$
joining two points $p$ and $q$ of $X$. We say that

  1. (i)

$q$ is one-sided conjugate to $p$ along $\gamma$ if there exists a sequence of
points $(q_{n})$ converging to $q$ such that for every $n$, there are two
distinct geodesics $\gamma_{n}^{1}$ and $\gamma^{2}_{n}$ joining $p$ to
$q_{n}$ and both converging to $\gamma$;

  2. (ii)

$p$ and $q$ are symmetrically conjugate along $\gamma$ if there exist
sequences of points $(p_{n})$ converging to $p$ and $(q_{n})$ converging to
$q$ such that that for every $n$, there are two distinct geodesics
$\gamma^{1}_{n}$ and $\gamma^{2}_{n}$ joining $p_{n}$ to $q_{n}$ and both
converging to $\gamma$;

  3. (iii)

$p$ and $q$ are unreachable conjugate along $\gamma$ if there are sequences of
points $(q_{n})$ converging to $q$ and $(p_{n})$ converging to $p$ such that
if $\gamma_{n}$ is geodesic joining $q_{n}$ to $p_{n}$ for all $n$, then the
sequence $(\gamma_{n})$ cannot converge to $\gamma$;

  4. (iv)

$p$ and $q$ are ultimate conjugate points along $\gamma$ if they are
symmetrically conjugate or unreachable conjugate along $\gamma$.

The relationship in sub-Riemannian geometry between these different
definitions of conjugate points is given by the following theorem.

######  Theorem 37.

Let $M$ be an ideal sub-Riemannian manifold, $\gamma:\interval{0}{1}\to M$ be
a normal geodesic such that its initial covector $\lambda_{0}$ is regular
conjugate and has finite order, and denote $p:=\gamma(0)$ and $q:=\gamma(1)$.
Then, the following statements are equivalent:

  1. (i)

$q$ is conjugate to $p$ along $\gamma$;

  2. (ii)

$q$ is one-sided conjugate to $p$ along $\gamma$;

  3. (iii)

$q$ is symmetrically conjugate to $p$ along $\gamma$.

Furthermore, if $p$ and $q$ are unreachable conjugate points along $\gamma$,
then $q$ is also conjugate to $p$ along $\gamma$.

######  Remark 38.

The ideal assumption in Theorem 37 is there to ensure that the sequences of
geodesics in Definition 36 consist of normal extremals.

###### Proof.

If $(q_{n})$ is a sequence of points converging to $q$ and if
$(\gamma_{n}^{1})$, $(\gamma^{2}_{n})$ are two sequences of geodesics joining
$p$ to $q_{n}$ and converging to $\gamma$, then their initial covectors
$\eta^{1}_{n}$ and $\eta^{2}_{n}$ will converge to $\lambda_{0}$ in
$\mathrm{T}^{*}_{p}(M)$. By Theorem 34, the normal extremals $\gamma^{1}_{n}$
and $\gamma_{n}^{2}$ must coincide for $n$ large enough. This proves that (i)
implies (ii). It is easy to see from Definition 36 that (ii) immediately
implies (iii).

Consider the function

$E:\mathrm{T}^{*}(M)\to M\times M:(x,\eta)\mapsto(x,\mathrm{exp}_{p}(\eta)),$

which has a differential at $(p,\lambda_{0})$ that is a linear isomorphism if
$q$ is not conjugate to $p$ along $\gamma$. By the inverse function theorem,
there exists a neighbourhood of $(p,\lambda_{0})$ on which the function $E$ is
a diffeomorphism. Suppose now that $q$ is one-sided conjugate to $p$ along
$\gamma$, so that there exists a sequence of points $(q_{n})$ converging to
$q$ and for all $n$ two distinct normal extremals $\gamma^{1}_{n}$ and
$\gamma^{2}_{n}$, with initial covectors $\eta^{1}_{n}$ and $\eta^{2}_{n}$
respectively, joining $p$ to $q_{n}$ and converging to $\gamma$. The proof of
(ii) implies (i) follows by observing that
$E(p,\eta^{1}_{n})=E(p,\eta^{2}_{n})$ for $n$ large enough while
$\eta^{1}_{n}\neq\eta^{2}_{n}$ since $\gamma_{n}^{1}$ and $\gamma^{2}_{n}$ are
distinct. By the same argument, it can be seen that (iii) implies (i). The
last statement about unreachable conjugate points is the same as in [SS09]. ∎

###  4.2 Structure of the sub-Riemannian cut locus

Theorem 1 is also related to the structure of the sub-Riemannian cut locus.
Let $M$ be an ideal sub-Riemannian manifold and define the cut time of
$(p,\lambda_{0})\in\mathrm{T}^{*}_{p}(M)$ as

$t_{\text{cut}}(p,\lambda_{0}):=\sup\\{t>0\mid\mathrm{exp}_{p}(\cdot\lambda_{0})|_{\interval{0}{t}}\text{
is a length-minimising geodesic}\\}.$

The (cotangent) cut locus is then

$\mathrm{Cut}(p):=\\{\lambda_{0}\in\mathrm{T}^{*}_{p}(M)\mid
t_{\text{cut}}(p,\lambda_{0})=1\\}.$

and we denote by $\mathrm{Cut}^{1}(p)$ the subset of $\mathrm{Cut}(p)$
consisting of those covectors $\lambda_{0}$ for which there exists another
$\lambda_{0}^{\prime}\in\mathrm{Cut}(p)$ such that
$\mathrm{exp}_{p}(\lambda_{0})=\mathrm{exp}_{p}(\lambda_{0}^{\prime})$ and
$\lambda_{0}\neq\lambda_{0}^{\prime}$. Note that by [ABB20, Theorem 8.72.], if
$\lambda_{0}\in\mathrm{Cut}(p)\setminus\mathrm{Cut}^{1}(p)$, then
$\lambda_{0}\in\mathrm{Conj}(p)$. In this specific case, we are able to prove
the local non-injectivity property of the exponential map without extra
assumptions and without invoking the regularity of the conjugate locus.

######  Theorem 39.

Let $M$ be an ideal sub-Riemannian manifold, and $p\in M$. If
$\lambda_{0}\in\mathrm{Cut}(p)\setminus\mathrm{Cut}^{1}(p)$, then the
exponential map $\mathrm{exp}_{p}$ fails to be injective in any neighbourhood
of $\lambda_{0}$.

###### Proof.

Let $\gamma(t):=\mathrm{exp}_{p}(t\lambda_{0})$ be the unique length
minimising geodesic between $p$ and $q:=\mathrm{exp}_{p}(\lambda_{0})$. Let
$u$ be its minimal control and $\lambda_{1}\in\mathrm{T}^{*}_{p}(M)$ its
Langrange multiplier. By [ABB20, Corollary 8.74.], we find a sequence of
points $q_{k}\in M$ converging to $q$ such that for each $k$ there are two
distinct length minimisers $\gamma_{k}^{1}$ and $\gamma_{k}^{2}$ joining $p$
and $q_{k}$. We denote by $\lambda_{1,k}^{1}$ and $\lambda^{2}_{1,k}$ the
normal Lagrange multipliers of $\gamma_{k}^{1}$ and $\gamma_{k}^{2}$
respectively, as well as $u_{k}^{1}$ and $u^{2}_{k}$ for their respective
minimal control. We write $\lambda_{0,k}^{1}$ and $\lambda_{0,k}^{2}$ for
their initial covectors. They satisfy
$\mathrm{exp}_{p}(\lambda_{0,k}^{1})=\mathrm{exp}_{p}(\lambda_{0,k}^{2})$ and
we would like to prove that $\lambda_{0,k}^{1}$ and $\lambda_{0,k}^{2}$ both
converge to $\lambda_{0}$ as $k\to+\infty$.

Modulo extraction of a subsequence, we may assume by compactness of length
minimisers ([ABB20, Proposition 8.67.]) that $u_{k}^{1}$ and $u^{2}_{k}$
converge in the strong $L^{2}$ topology, as well as $\gamma_{k}^{1}$ and
$\gamma_{k}^{2}$ converge uniformly to a geodesic joining $p$ and $q$. This is
unique by our assumption. So, $\gamma_{k}^{1}$ and $\gamma_{k}^{2}$ converge
uniformly to $\gamma$ and $u_{k}^{1}$ and $u^{2}_{k}$ converge to $u$.

Choose any metric $|\cdot|$ on $\mathrm{T}^{*}_{p}(M)$ that we only use to
prove estimates. We want to show that $\lambda_{1,k}^{1}$ is convergent.
Assume that it is not, this would mean that there exists a subsequence of
$|\lambda_{1,k}^{1}|$ that diverges to $+\infty$. Now,
$\eta_{1,k}^{1}:=\lambda_{1,k}^{1}/|\lambda_{1,k}^{1}|$ of course converges to
some $\eta_{1}^{1}$. The Lagrange multiplier rule ([ABB20, Section 8.3]) for
$\gamma^{1}_{k}$ implies that

$\frac{\lambda_{1,k}^{1}}{|\lambda_{1,k}^{1}|}D_{u_{k}}E_{p}=\frac{u_{k}}{|\lambda_{1,k}^{1}|}$

and thus, by taking the limit, $\eta_{1}^{1}D_{u}E_{p}=0$. Here, we have
written $E_{p}$ for the endpoint map (see [ABB20, Section 8.1]). We obtain
that $\eta_{1}^{1}$ is an abnormal Lagrange multiplier but this is impossible
since we assumed $\gamma$ to be non-abnormal.

Therefore, $\lambda_{1,k}^{1}$ must be convergent and by taking again the
limit in the Lagrange multiplier rule

$\lambda_{1,k}^{1}D_{u_{k}}E_{p}=u_{k}$

we obtain that its limit must a (normal) Lagrange multiplier for $\gamma$.
This is necessarily unique since $\gamma$ is not abnormal and thus
$\lambda_{1,k}^{1}\to\lambda_{1}$, as well as
$\lambda_{1,k}^{2}\to\lambda_{1}$ by the same argument. This also implies that
$\lambda_{0,k}^{1}\to\lambda_{0}$ and $\lambda_{0,k}^{2}\to\lambda_{0}$ since
initial and final covectors are linked by the Hamiltonian flow. ∎

######  Corollary 40.

Let $M$ be an ideal sub-Riemannian manifold, and $p\in M$. The set
$\mathrm{Cut}^{1}(p)$ is dense in $\mathrm{Cut}(p)$.

###### Proof.

Let $\lambda_{0}\in\mathrm{Cut}(p)\setminus\mathrm{Cut}^{1}(p)$ and consider a
sequence of decreasing open neighbourhoods
$(U_{n})\subseteq\mathrm{T}^{*}_{p}(M)$ such that
$\cap_{n}U_{n}=\\{\lambda_{0}\\}$. Then, by Theorem 34, the exponential map
$\mathrm{exp}_{p}$ fails to be injective on each $U_{n}$ and so for every $n$,
the intersection $\mathrm{Cut}^{1}(p)\cap U_{n}$ is non emptry. We therefore
find a sequence of covectors $\lambda_{n}\in\mathrm{Cut}^{1}(p)$ such that
$\lambda_{n}\to\lambda_{0}$. ∎

Corollary 40 can be used to prove that the map $\mathrm{d}(p,\cdot)^{2}$ is
smooth in a neighbourhood of $q$ if and only if there is a unique length-
minimiser $\gamma$ joining $p$ to $q$ and $q$ is not conjugate to $p$ along
$\gamma$. In the non-ideal case, one may bypass this type of argument to study
the regularity of the sub-Riemannian squared distance (see [ABB20, Chapter
11]).

###  4.3 Final remarks

  1. 1)

[Kli95] gives an inconclusive argument towards non-local injectivity of the
Riemannian exponential map at critical points. Namely [Kli95, 2.1.12 Theorem]
argues indirectly, assuming that $\mathrm{exp}_{p}$ is injective in a
neighbourhood of a conjugate vector $v$ (and thus a homeomorphism by
invariance of domain). A curve $b$ that has the same endpoints as the geodesic
$c$ along which $p$ is conjugate to $\mathrm{exp}_{p}(v)$ is constructed so
that one may use standard arguments using Jacobi fields to conclude that
$\mathrm{L}(b)<\mathrm{L}(c)$. The final step consists in defining the curve
$\tilde{b}:=\mathrm{exp}_{p}^{-1}(b)$ in $\mathrm{T}_{p}(M)$ and to invoke a
corollary of Gauss’ lemma from [Kli95, 1.9.2 Theorem] to conclude that
$\mathrm{L}(b)\geq\mathrm{L}(c)$. However, this very last step can only be
performed if $\tilde{b}$ is (piecewise) smooth. However, $\mathrm{exp}_{p}$
needs to be only a homeomorphism. This is in fact the heart of the problem
that needs to be overcome, from the orginal arguments of Morse - Littauer, and
Savage ([ML32], [Sav43]), to the one of Warner ([War65]), and indeed the
present work.

  2. 2)

Even if the sub-Riemannian manifold $M$ is ideal, it seems possible that a
geodesic has a conjugate point of infinite order. Finding a relevant example
could help to fully understand the structure of the conjugate locus. Is it
possible to construct an example for which $\mathrm{Conj}(p)$ contains only
conjugate covectors of infinite order?

  3. 3)

The proof of Theorem 34 relies on the submanifold structure of
$\mathrm{Conj}(p)$ near a regular conjugate covector of finite order. In the
case of infinite order, can we still find such a manifold structure?

  4. 4)

Warner’s proof of Theorem 34 is different to what we have done here. At the
core of Warner’s proof is Whitney’s singularity theorem to find the normal
forms of the exponential map near a conjugate vector. What are the normal
forms of the sub-Riemannian exponential map? Due to the reliance of this
argument on the order of the conjugate value, there might not be a general
answer to that question. Rather each class of examples could be investigated.
We have pursued this approach for the Heisenberg group in [BK21] and we expect
it to be successful in other cases, possibly in a similar way to what has been
done for the 3D contact case in [ABB20, Chapters 17 to 19] (and the references
therein). In [BBCN17], the authors classify the singularities of the sub-
Riemannian exponential for low-dimensional generic structures.

  5. 5)

The study of conjugate points along abnormal geodesics is completely open.

## References

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