Regularity and Continuity Properties of the Sub-Riemannian Exponential Map

We prove a version of Warner’s regularity and continuity properties for the sub-Riemannian exponential map. The regularity property is established by considering sub-Riemannian Jacobi fields while the continuity property follows from studying the Maslov index of Jacobi curves. We finally show how this implies that the exponential map of the three-dimensional Heisenberg group is not injective in any neighbourhood of a conjugate vector.


Introduction
In his work [War65] on the conjugate locus in Riemannian geometry, Warner introduced the notion of regular exponential map, a map F : T q (M) → M, where M is a finite dimensional smooth manifold, that satisfies three conditions: a non-vanishing speed condition along rays, a regularity and a continuity condition.Furthermore, Warner shows that such a map is non-injective in any neighbourhood of any singularities of F .This is done through studying the normal forms of F around singularities, namely the points where the jacobian determinant of F vanish.Warner then proves that the exponential map of a Finsler manifold is regular in this sense, giving an alternative proof of a result due to Morse and Littauer [ML32].Warner's theorem was adapted to Lorentzian structures in [Ros83] and then to semi-Riemannian manifolds in [Sze08].
In the present work, we adapt Warner's conditions for the exponential map in sub-Riemannian geometry.Because of the lack of a Levi-Civita connection, the study of geodesics is carried out from the Hamiltonian viewpoint (see Section 2 for a summary of the theory).Length minimisers are found to be normal and/or abnormal.Normal geodesics are solutions of a Hamiltonian system of differential equations with initial conditions taking values in the cotangent space.The sub-Riemannian exponential map is the projection of the corresponding Hamiltonian flow.Abnormal geodesics can also appear: they are length minimisers that satisfy a condition not characterised by a differential equation.Strongly normal geodesics are those that do not contain abnormal subsegments.
Let us introduce some notations to state our main results.The normal geodesic starting at q ∈ M with initial covector λ 0 ∈ T * q (M) is denoted by γ q,λ 0 and I q,λ 0 is its maximal domain.The ray through λ 0 is the map defined by r q,λ 0 (t) := tλ 0 for t ∈ I q,λ 0 .We denote by H q the restriction of the Hamiltonian to a fiber T * q (M).For A ∈ Ker(d λ 0 exp q ), the sub-Riemannian Jacobi field J A along γ q,λ 0 is the one with initial values (0, A).Choosing a symplectic moving frame along γ q,λ 0 allows us to introduce ∇J A , a (non canonical) derivative of J A along the curve.The theory of sub-Riemannian Jacobi fields will be detailled in Section 3.2.
Theorem 1 (Regularity of the sub-Riemannian exponential map).Let M be a sub-Riemannian manifold, q ∈ M and exp q : A q → M be the corresponding exponential map with domain A q ⊆ T * q (M).Then, (R1) The map exp q is C ∞ on A q and, for all λ 0 ∈ A q \ H −1 q (0) and all t ∈ I q,λ 0 , we have d tλ 0 exp q (ṙ q,λ 0 (t)) = 0 exp q (tλ 0 ) ; (R2) For every λ 0 ∈ A q \ {0} and every symplectic moving frame along the cotangent lift λ(t) of the normal geodesic γ(t) := exp q (tλ 0 ), the map Ker(d λ 0 exp q ) → T exp q (λ 0 ) (M)/ d λ 0 exp q (T λ 0 (T * q (M))), sending A to ∇J A (1) + d λ 0 exp q (T v (T * q (M))), is a linear isomorphism; (R3) Let λ 0 ∈ A q \ H −1 q (0) be a covector such that the corresponding geodesic γ(t) := exp q (tλ 0 ) is strongly normal.Then, there exists a radially convex neighbourhood V of λ 0 such that for every ray r q,λ 0 which intersects V that does not contain abnormal subsegments in V, the number of singularities of exp q (counted with multiplicities) on Im(r q,λ 0 ) ∩ V is constant and equals the order of λ 0 as a singularity of exp q , i.e. dim(Ker(d λ 0 exp q )).
Condition (R1) follows from the constant speed property of normal geodesics, see Section 3.1.The rich theory of sub-Riemannian Jacobi fields, developed in [BR17] for example, will help us to prove condition (R2) in Section 3.2.In Riemannian geometry, condition (R3) is a consequence of Morse's index theory.His ideas are adapted to this context with the Maslov index and the condition (R3) will be obtained in Section 3.3.
Warner uses these three conditions in [War65] to conclude that the Riemannian exponential map is not locally injective around a singularity.This result is originally due to Morse and Littauer [ML32].In this paper, Warner's method is used for the first time in sub-Riemannian geometry, in the case of the three dimensional Heisenberg group H in Section 4. We expect it to work for larger classes of sub-Riemannian manifolds as well.
Theorem 2. For q ∈ H, the sub-Riemannian exponential map exp q : A q → H is not injective on any neighbourhood of a conjugate vector λ 0 ∈ A q \ H −1 q (0).

Acknowledgements
This work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/N509462/1 (1888382) for Durham University.
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).

Preliminaries
In this section, we set up the basics of sub-Riemannian geometry.We rely on [ABB20] and [Agr08] for the general theory.
2.1.Sub-Riemannian geometry.We begin with a definition of sub-Riemannian structure.
The family D of horizontal vector fields is defined as The rank of the sub-Riemannian structure at q ∈ M is rank(q) := dim(D q ).Observe that in our definition, a sub-Riemannian manifold can be rank-varying.Definition 4. We say that curve γ : [0, T ] → M is horizontal if γ is Lipschitz in charts and if there exists u ∈ L 2 ([0, T ], E) such that for almost every t ∈ [0, T ], we have u(t) ∈ E γ(t) and γ(t) = f E (u(t)).The sub-Riemannian length and the sub-Riemannian energy of γ are defined by where v Dq := min u, u Eq | u ∈ E q and f E (u) = (q, v) for v ∈ D q and q ∈ M.
Remark 5.The norm • Dq is well defined, induced by an inner product •, • Dq via the polarisation formula and the map t → γ(t) D γ(t) is measurable.
In the case where every two points can be joined by a horizontal curve, we have a well defined distance function on M. Definition 6.The distance of a sub-Riemannian manifold M, also called the Carnot-Carathéodory distance, is defined by In this work, we assume that the sub-Riemannian structures satisfy the Hörmander condition, that is to say Lie q (D) = T q (M) for all q ∈ M. We also say that D is bracketgenerating.This is motivated by the following well-known result.
Theorem 7 (Chow-Rashevskii theorem, see [ABB20,Theorem 3.31.]).Let M be a sub-Riemannian manifold such that its distribution D is C ∞ and satisfies the Hörmander condition.Then, (M, d) is a metric space and the manifold and metric topology of M coincide.
Given m global vector fields X 1 , . . ., X m : M → T(M) on a manifold M, we can build on it a sub-Riemannian structure in the following way.We set and finally the metric on E is the Euclidean one.In this way, we induce an inner product on D q = span{X 1 (q), ..., X m (q)} by the polarization formula applied to the norm The family (X 1 , . . ., X m ) is said to be a generating family of the sub-Riemannian manifold.
A free sub-Riemannian structure is one that is induced from a generating family.Every sub-Riemannian structure is equivalent to a free one (see [ABB20, Section 3.1.4]).From now on, we will therefore assume, without loss of generality, that every sub-Riemannian manifold is free.
2.2.End-point map and Lagrange multipliers.We fix a sub-Riemannian manifold M for which the family (X 1 , . . ., X m ) is generating.From an optimal control point of view, a curve γ : In fact, by Carathéodory's theorem for ordinary differential equations, we know that there exists a unique maximal Lipschitz solution to the Cauchy problem for every u ∈ L 2 ([0, T ], R m ), q ∈ M and t 0 ∈ [0, T ].We denote such a solution by γ t 0 ,q,u and we can now introduce the end-point map.
Definition 8. Let q ∈ M and T > 0. The end-point map at time T > 0 is the smooth map E q,T : U → M : u → γ 0,q,u (T ), where is the open subset of controls such that γ 0,q,u , solution to the Cauchy problem (2), is defined on the whole interval [0, T ].
On the space of controls L 2 ([0, T ], R m ), we can define a length functional, as well as a corresponding energy functional Given an horizontal curve γ : [0, T ] → M, we define at every differentiability point of γ the minimal control u associated with γ In particular, the previous functionals are related to the sub-Riemannian length and energy: L(γ) = L(u) and J(γ) = J(u), where u is the minimal control associated with γ.
Through the Cauchy problem (2), it is clear that finding a length minimiser for L among the horizontal curves with fixed end-points γ(0) = q and γ(T ) = q ′ is equivalent to finding a minimal control for L for which the associated path joins q and q ′ .Furthermore, we have the following classical correspondence: a horizontal curve γ : [0, T ] → M joining q to q ′ is a minimiser of J if and only if it is a minimiser of L and is parametrised by constant speed.
In terms of the end-point map, the problem of finding the minimisers joining two fixed point q, q ′ ∈ M is thus equivalent to solving the constrained variational problem min J(u) u ∈ E −1 q,T (q ′ ) . (3) The Lagrange multipliers rule provides a necessary condition to be satisfied by a control u that is a constrained critical point for (3).
Proposition 9 (Lagrange Multipliers).Let u ∈ U be an optimal control for the variation problem (3).Then at least one of the following statements holds true (i) there exists λ(T An optimal control is called normal (resp.abnormal) when it satisfies the condition (i) (resp.(ii)).The same terminology is used for the corresponding curve γ u .We note that the extremal trajectory γ u could be both normal and abnormal.A normal trajectory γ : [0, T ] → M is called strictly normal if it is not abnormal.If, in addition, the restriction γ| [0,s] is strictly normal for every s > 0, we say that γ is strongly normal.It can be seen that γ is strongly normal if and only if the normal geodesic γ does not contain any abnormal segment.2.3.Characterisation of sub-Riemannian geodesics.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 can not have a geodesic equation through a covariant derivative.Rather, sub-Riemannian geodesic are characterised via Hamilton's equations.
We recall that the Hamiltonian vector field of a map a ∈ C ∞ (T * (M)) is the unique vector field − → a on T * (M) that satisfies where ω denotes the canonical symplectic form on the cotangent bundle T * (M).
The smooth control-dependent Hamiltonian of a sub-Riemannian structure is the map h : R m × T * (M) → R defined as It is easy to see that, by strict convexity, there exists a unique maximum u(λ) of u → h u (λ) for every λ ∈ T * (M).Therefore, a maximized Hamiltonian, or simply Hamiltonian, is well defined . Furthermore, the Hamiltonian H can written in terms of the generating family of the sub-Riemannian structure (X 1 , . . ., X m ), as follows where λ := (q, λ 0 ) ∈ T * (M) and h k (q, λ 0 ) := λ 0 , X k (q) .For q ∈ M, we will also write H q for the restriction of H to the cotangent space T * q (M).The Lagrange multipliers rule may be further developed to characterise normal extremals as curves that satisfy Hamilton's differential equation.Alternatively, the following result can also be seen as an application of the Pontryagin Maximum Principle to the sub-Riemannian length minimisation problem.
Theorem 10 (Pontryagin's Maximum Principle).Let γ : [0, T ] → M be a horizontal curve which is a length minimiser among horizontal curves, and parametrised by constant speed.Then, there exists a Lipschitz curve λ(t) ∈ T * γ(t) (M) such that one and only one of the following is satisfied: ( If λ satisfies (N) (resp.(A)), we will also say that λ is a normal extremal (resp.abnormal extremal).The projection of a normal extremal to M is locally minimising, that is to say it is a normal geodesic parametrised by constant-speed.This can be seen of an application of the invariance of the Hilbert integral around small subsegments of the trajectory, defined through the Poincarré-Cartan one-form (see [ABB20, Section 4.7]).However, the projection of an abnormal extremal to M might not be locally minimising.
It does happen that a sub-Riemannian structure does not have any non-trivial (i.e.non-constant) abnormal geodesic (the trivial geodesic is always abnormal as soon as rank(D q ) < dim(M)).In this case, the sub-Riemannian manifold is said to be ideal.
The theory of ordinary differential equations provides the existence of a maximal solution to (N) for every given an initial condition (q, λ 0 ) ∈ T * (M).The flow of Hamilton's equation is denoted by e t − → H .We finally turn our attention to the central object of this work: the sub-Riemannian exponential map.
Definition 11.The sub-Riemannian exponential map at q ∈ M is the map is the open set of covectors such that the corresponding solution of (N) is defined on the whole interval [0, 1].
The sub-Riemannian exponential map exp q is smooth.If λ : [0, T ] → T * (M) is the normal extremal that satisfies the initial condition (q, λ 0 ) ∈ T * (M), then the corresponding normal extremal path γ(t) = π(λ(t)) by definition satisfies γ(t) = exp q (tλ 0 ) for all t ∈ [0, T ].If M is complete for the Carathéodory distance, then A q = T * q (M) and if in addition there are no stricly abnormal length minimisers, the exponential map exp q is surjective.Contrary to the Riemannian case, the sub-Riemannian exponential map is not necessarily a diffeomorphism of a small ball in T * q (M) onto a small geodesic ball in M. In fact, Im(d 0 exp q ) = D q and exp q is a local diffeomorphism at 0 ∈ T * q (M) if and only if D q = T * q (M).Our aim is to prove that exp q is regular in the sense of Warner [War65].

Regularity and Continuity of the sub-Riemannian exponential map
3.1.Normal extremals.As pointed out in the previous section, the normal geodesic γ(t) of the sub-Riemannian manifold M, with initial point γ(0) = p and initial covector λ 0 ∈ A q is the projection of the normal extremal λ : I q,λ 0 → T * (M), the solution to Hamilton's geodesic equation with initial value (q, λ 0 ).The ray in A q through λ 0 is the map r q,λ 0 : I q,λ 0 → T * q (M) : t → tλ 0 where I q,λ 0 ⊆ R + is the maximal interval containing 0 such that tλ 0 ∈ A q for every t ∈ I q,λ 0 .In this way, ṙq,λ 0 (t) ∈ T tλ 0 (T * q (M)) and identifying T tλ 0 (T * q (M)) with T * q (M) in the usual way, we have ṙq,λ 0 (t) = λ 0 for every t ∈ I q,λ 0 .
Proposition 12 (see [ABB20], Theorem 4.25).Let λ : [0, T ] → T * (M) be a normal extremal, that is a solution to Hamilton's equation λ = − → H (λ). The corresponding normal geodesic γ(t) = π(λ(t)) has constant speed and 1 2 γ(t) 2 = H(λ(t)) for every t ∈ [0, T ].Proof.The Hamiltonian is constant along a normal trajectory: The minimal control for the curve γ = π • λ is given by u In view of this result, we observe that, contrary to the Riemannian case, there might exist initial covectors λ 0 ∈ T * q (M) such that the corresponding normal geodesic is trivial.This can happen if λ 0 ∈ H −1 q (0).Since the normal geodesic γ has constant-speed, we have which is non-zero as long as λ 0 ∈ A q \ H −1 q (0).This proves a cotangent version of the first condition of Warner.
Theorem 13 (Constant speed property).The map exp q is C ∞ on A q and, for all λ 0 ∈ A q \ H −1 q (0) and all t ∈ I q,λ 0 , we have d tλ 0 exp q (ṙ q,λ 0 (t)) = 0 exp q (tλ 0 ) .The set A q \ H −1 (0) is open and radially convex in the following sense: a subset 3.2.Jacobi fields and the regularity property.As will be shown, this property, called regularity property, is a feature of Jacobi fields, which theory we outline here in the sub-Riemannian context.
Let γ : [0, T ] → M be a normal geodesic and λ(t) be its cotangent lift.We can write γ(t) = exp q (tλ 0 ) for some initial covector λ 0 ∈ T * q (M).Consider a variation of γ(t) through normal geodesics Γ(t, s) = exp σ(s) (tV (s)) where Λ(s) = (σ(s), V (s)) is a curve in T * (M) with Λ(0) = (q, λ 0 ).The curve Λ is well defined on a small interval (−ε, ε).A sub-Riemannian Jacobi field J along the normal geodesic γ can be seen as the variation field of a variation γ through normal geodesics: Remembering that exp q (tv) = π • e t − → H (q, v), we have the equalities The Jacobi field J along γ is therefore uniquely determined by its initial value Λ(0) ∈ T (q,λ 0 ) (T * (M)).This implies that the space of Jacobi fields along the geodesic γ, which we denote by J (γ), is a vector space of dimension 2n.On the other hand, the space of Jacobi fields along the extremal λ, denoted this time by J (λ) is the collection of vector fields along λ of the form also uniquely determined by Λ(0) ∈ T (q,λ 0 ) (T * (M)).The space J (γ) is linearly isomorphic to J (λ) through the pushforward of the bundle projection π : T * (M) → M. Equivalently, a vector field J is a Jacobi field along the extremal λ if it satisfies where L − → H J is the Lie derivative of a vector field along λ in the direction of − → H : The equation (4) can be rewritten using the symplectic structure of T * (M) and moving frame generalizing, in a non-canonical way, the Riemannian parallel transport (see also [BR17] for more details).
Theorem 14.Let γ : [0, T ] → M be a normal geodesic and λ(t) its cotangent lift.There exists a frame Furthermore, given such a moving frame, a vector field ) is a Jacobi field along λ if and only the following Hamilton's equation for Jacobi fields is satisfied: for some matrix A(t) with rankA(t) = dim D γ(t) and some symmetric matrices B(t), R(t).
Definition 15.The derivative of a Jacobi field J along γ with coordinates (p, x) is defined as The next two results are at the heart of the second condition of Warner.
Lemma 16.Let γ : [0, T ] → M be a normal geodesic.Suppose that J and J are two Jacobi fields along γ.Once we fix a symplectic moving frame E 1 (t), . . ., E n (t), F 1 (t), . . ., F n (t) along the cotangent lift λ(t) as given by Theorem 14, we have that Remark 17.This result is a generalisation of a well-known fact in Riemannian geometry: If J 1 and J 2 be two Riemannian Jacobi fields along a geodesic γ, then D t J 1 , J 2 − J 1 , D t J 2 is constant along γ, where D t stands for the covariant derivative.
Proof.Let (p, x) (resp.(p, x)) be the coordinates of the Jacobi field J (resp.J) with respect to the moving frame.Hamilton's equation for Jacobi fields (5) states that ṗ(t) = −A(t) T p(t) + R(t)x(t) and ẋ(t) = B(t)p(t) + A(t)x(t) and since R(t) and B(t) are symmetric, we have This holds for every t ∈ [0, T ] which concludes the proof.
Let γ : I → M be a normal geodesic and a, b ∈ I.We use the notation J a (γ) for the vector space of Jacobi fields along γ vanishing at time t = a and J a,b (γ) for the subspace of J a (γ) of Jacobi fields along γ : I → M vanishing at both t = a and t = b.Proposition 18.Let γ : [0, T ] → M be a normal geodesic with initial covector λ 0 ∈ T * q (M) and such that γ(0) = q ∈ M. Fix also a symplectic moving frame as provided by Theorem 14.Then, the sets Proof.Choose a basis (J 1 , . . ., J k ) of J 0,t (γ) and complete it into a basis for J 0 (γ) with (J 1 , . . ., J n−k ).
The family (∇J 1 (t), . . ., ∇J k (t)) is linearly independent.Indeed, assume this is not the case, then there exists a 1 , . . ., a k ∈ R such that k i=1 a i ∇J i (t) = 0.The Jacobi field J := k i=1 a i J i satisfies J(0) = 0, J(t) = 0 and ∇J(t) = 0.By Hamilton's equation for Jacobi fields, we can conclude that J is identically zero and therefore Similarly, the family (J 1 , . . ., J n−k ) is linearly independent.If this is not the case, then there exists b We have J(0) = 0 and J(t) = 0. So, the Jacobi field J ∈ J 0,t (γ) and hence b Finally, Lemma 16 implies that It remains to link Jacobi fields and the exponential map.The following two results are analogous to the Riemannian context.Firstly we examine Jacobi fields along γ vanishing at its initial time.
Secondly, we look at the singularities of the exponential map.These are covectors λ 0 ∈ A q such that Ker(d λ 0 exp q ) is not trivial.These covectors are called conjugate (co)vectors and the point exp q (λ 0 ) is said to be conjugate to p. Proposition 20.Let γ : [0, 1] → M be a normal geodesic with initial covector λ 0 ∈ A q and such that γ(0) = q ∈ M. The covector λ 0 is a critical point for exp q if and only if there exists a non trivial Jacobi field J along γ such that J(0) = 0 and J(1) = 0.
Proof.If λ 0 is a singularity of exp q , there exists a vector λ 0 such that d λ 0 exp q (λ 0 ) = 0.In that case, from the previous proposition, the vector field is a non trivial Jacobi field field such that J(0) = 0 and J(1) = 0.The converse implication is similar.
For A ∈ Ker(d λ 0 exp q ), we denote by J A (t) ∈ J 0,1 (γ) the Jacobi field along γ with initial value (0, A).We finish this section by proving the following cotangent version of Warner's second condition.
Remark 22.In light of Proposition 18, Proposition 20, Proposition 21 and its proof, we can see that A γ(1) = d λ 0 exp q (T λ 0 (T * q (M))) and In particular, the subspace B γ(1) does not depend on the choice of moving frame along λ(t).
Let us describe the results of this section in a different way.The introduction of a moving frame as given in Theorem 14 is equivalent to choosing a horizontal complement of some Riemannian metric g on M. Indeed, given a moving frame E 1 (t), . . ., E n (t), F 1 (t), . . ., F n (t), we can naturally obtain a Riemannian metric g that extends •, • γ(t) along the curve γ(t) to the whole manifold M. Conversely, choosing a Riemannian metric g on M induces an isomorphism: where ξ ♯ is the unique element of T q (M) such that g(ξ ♯ , X) = ξ(X), for every X ∈ T q (M), the spaces Ver λ and T * q (M) being canonically identified.Now, if X ∈ T q (M) and if Therefore, from an orthonormal basis X 1 , . . ., X n along γ(t), we can construct the moving frame E 1 (t), . . ., E n (t), F 1 (t), . . ., F n (t) by setting E ♯ i (t) = X i (t) and X i (t) = d λ(t) π(F i (t)), and Theorem 14 would follow from Equation (6).Different choices of g therefore corresponds to different choices of Darboux moving frames.
When such a Riemannian metric g on M has been fixed, the derivative of a Jacobi field along γ with initial value ξ 0 corresponds to Finally, Proposition 18 is saying that the map ♯ has its image g-perpendicular to Im(d λ 0 exp q ), leading to Proposition 21 3.3.Maslov index and the continuity property.We now approach Jacobi fields and conjugate points via Lagrange Grassmannian geometry.For the definitions and properties related to Jacobi curves, we refer the reader to [ABB20, Chapter 15], while the Maslov index, in its full generality, is developed in [PT08, Chapter 5], for example.
We start with Lagrange Grassmannian geometry.Let (Σ, ω) be a symplectic vector space of dimension 2n.A Lagrangian subspace of Σ is a subspace of dimension n on which the restriction of ω vanishes.The collection of all Lagrangian subspaces of Σ is called the Lagrange Grassmannian of Σ, and is denoted by L(Σ).It is a compact manifold of dimension n(n + 1)/2.Furthermore, if Λ ∈ L(Σ), there is a linear isomorphism between the tangent space T Λ (L(Σ)) and the space of bilinear forms Q(Λ) on Λ.The linear isomorphism is given by where Λ(z) := ω(z(0), ż(0)), where we consider a smooth curve Λ(t) in L(Σ) such that Λ(0) = Λ and a smooth extension z(t) ∈ Λ(t) such that z(0) = 0.It can be shown that Λ is a well-defined quadratic map, independent of the extension considered.

Definition 24. An isomorphism
is defined by requiring that any positive generator of is a continuous curve with endpoints in Λ 0 (L 0 ), we denote by µ L 0 (Λ(•)) ∈ Z the integer number that corresponds to the homology class of Λ(•) by the isomorphism (7).The number µ L 0 (Λ(•)) is called the Maslov index of the curve Λ(•) relative to the Lagrangian L 0 .
The key properties of the Maslov index, including the fact that it is homotopy invariant, is summarized in the next theorem.
Theorem 25 ([PT08, Lemma 5.1.13and Corollary 5.1.18]).Let Λ : [a, b] → L(Σ) be a curve with endpoints in Λ 0 (L 0 ).We have , where ⋆ denotes the concatenation of curves; i.e. there is exists a continuous function where sgn(B) is the signature of B, that is to say, Remark 26.Theorem 25 (v) states the homotopy invariance of the Maslov index.This property was firstly proved in [Arn67].
We now turn our attention to the concept of Jacobi curve and the continuity property.As seen in the previous section, a conjugate vector occurs when there is a non trivial Jacobi field along the geodesic that vanishes both at its initial and endpoint.It seems therefore natural to study the evolution of the space of all Jacobi fields at a time t that vanish at its initial time: for q ∈ M and λ 0 ∈ A q .For every t ∈ [0, 1], the set L (q,λ 0 ) (t) is a Lagrangian subspace of T λ(t) (T * (M)).In order to be able to use the geometry and theory of Lagrangian Grassmannian, we will work with an alternative curve that lives in the fixed Grassmannian T (q,λ 0 ) (T * (M)).
The relation between the Jacobi curve and the conjugates vectors is given in next proposition.
In other words, sλ 0 is conjugate if and only if the Jacobi curve J (q,λ 0 ) (t) is in Λ 1 (L 0 ) for t = s.Proposition 30 alongside the condition of abnormality for geodesics shows that if we have a segment of points that are conjugate to the initial one, then the segment is also abnormal.curve J λ (t) with endpoints transverse to J λ (0) is equal to the number of intersection (counted with multiplicity, by Proposition 32), and since such intersections corresponds to conjugate times (the dimension of the intersection being the multiplicity, by Proposition 30 and Corollary 31).The truncated cone over W given by V := τ λ 0 | λ 0 ∈ W, τ ∈ [t − , t + ] is the required radially convex set given by the continuity property.
Remark 35.In fact, even if λ 0 corresponds to a normal geodesic with abnormal subsegments in V, the proof of Proposition 33 says that J (q,λ 0 ) | [t − ,t + ] and J (q,λ 0 ) | [t − ,t + ] have the same Maslov index with respect to J (q,λ 0 ) (0) and J (q,λ 0 ) (0).This also completes the proof of Theorem 1.It is reasonable to ask whether the cotangent version of Warner's regularity conditions implies a sub-Riemannian analogue of Morse-Littauer's theorem, i.e. the non-injectivity of the sub-Riemannian exponential map on any neighbourhood of a conjugate covector.Warner's approach involves giving the normal forms of the exponential map on neighbourhood of (regular) conjugate vectors.It is not obvious to us that Theorem 1 would easily provide such a local description of the sub-Riemannian exponential map about its singularities.However, we are able to pursue Warner's program for a specific example: the three dimensional Heisenberg group.

Exponential map of the Heisenberg group
In this section, we study the three dimensional Heisenberg group and prove Theorem 2. 4.1.Geometry of the Heisenberg group.The Heisenberg group H is the sub-Riemannian structure, defined on R 3 , that is generated by the global vector fields The Heisenberg group H enjoys a structure of Lie group when equipped with the law where z := (x, y) and z ′ := (x ′ , y ′ ) are elements of R 2 that we will identify with C for convenience (• denotes the complex conjugation).The neutral element of this operation is (0, 0, 0) and the inverse of (x, y, τ ) is (−x, −y, −τ ).There are many good references on the Heisenberg group, see [CDPT07] or [ABB20] for example.
The Hamiltonian H : T * (H) → R is thus given by for λ = (q, udx| q + vdy| q + αdτ | q ) and q = (x, y, τ ).In complex coordinates, we will write z := x + iy and w := u + iv.Since the sub-Riemannian structure is left-invariant, we may choose q = (0, 0, 0) for the rest of this section, without loss of generality.We can explicitly solve Hamilton's equations to find the expression of a normal geodesic starting from' q = (0, 0, 0) of H with initial covector λ 0 = u 0 dx| q + v 0 dy| q + α 0 dτ | q ∈ T * q (H): The symplectic moving frame (E induced by the global coordinates of H n satisfies Theorem 14.The Jacobi fields along γ are thus determined by the differential equation A computation in the Heisenberg group with the chosen symplectic frame shows that the block matrices in the above differential equation are given by and the symmetric matrix Solving this ordinary differential equation yields the general form of a Jacobi field J (t) = n i=1 p i (t)E i (t) + x i (t)F i (t) along λ(t) with initial condition (p(0), x(0)).After some calculations, we find where With this explicit description of sub-Riemannian Jacobi fields along normal geodesics of H, we are now able to study conjugates vectors.Alternatively, this could also be done by computing the determinant of the sub-Riemannian exponential map.We use the Jacobi fields characterisation of the kernel of exp q to illustrate our work.
Proposition 36.The covector λ 0 = u 0 dx| q + v 0 dy| q + α 0 dτ | q ∈ T * q (H) is a conjugate covector of q = (0, 0, 0) ∈ H with H(λ 0 ) = 0 if and only if α 0 sin(α 0 )+2 cos(α 0 )−2 = 0 and α 0 = 0. Furthermore, the conjugate vectors are all of order one, they form a submanifold of T * q (H) of dimension 2 and Proof.We have seen in Section 3.2 that A covector λ 0 will be conjugate if the kernel of the 3×3 bottom left block matrix of M(1) is non trivial.Furthermore, the image of those vectors under M 1 (1) will give Ker(d λ 0 exp q ).If α 0 tends to 0, the block is similar to Since we assume that H(λ 0 ) = 0, the matrix above has a trivial kernel and α 0 = 0 does not produce a conjugate vector.

Local non injectivity of the exponential map of the Heisenberg group.
The set of all conjugate covectors to q is called the conjugate locus at q and we denote it by Conj q (H).
The structure of the conjugate locus being established, we can finally prove Morse-Littauer's theorem for the Heisenberg group by a novel application of Warner's approach from [War65, Theorem 3.4.] in sub-Riemannian geometry.We expect this method to work in other classes of sub-Riemannian manifolds.
Proof of Theorem 2. Let λ 0 ∈ A q \ H −1 q (0) be a conjugate vector of q ∈ H.By Proposition 36, we can thus choose a two-dimensional open connected submanifold C of Conj q (H) in T * q (H) containing λ 0 .In particular, the conjugate vectors in C have all order 1.We also write C 0 (resp.C 1 ) for the set of covectors λ 0 ∈ C such that dim Ker(d λ 0 exp q ) ∩ T λ 0 (Conj q (H)) = 0 (resp.= 1).
Case 1 : λ 0 ∈ C 1 .The subspaces Ker(d λ 0 exp q ) with λ 0 ∈ C 1 form a one-dimensional and involutive smooth distribution of C 1 .This is because it corresponds to the distribution induced by the kernels of d λ 0 exp q .By Frobenius theorem, there exists a unique integral manifold passing through λ 0 .This is a one dimensional connected submanifold N of C 1 such that T λ 0 (N) = Ker(d λ 0 exp q ) for all λ 0 ∈ N. We then have that the restriction of exp q to N satisfies d λ 0 (exp q | N ) = 0 for every λ 0 ∈ N. Since N is connected, this implies that the sub-Riemannian exponential map maps every elements of N into a single point and hence exp q is not injective in any neighbourhood of λ 0 ∈ C 1 .