Regularity and Continuity properties of the sub-Riemannian exponential map

Abstract

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.

Local non-injectivity of the exponential map at critical points in sub-Riemannian geometry

“…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 [BK21]. This allows us to define regular/singular conjugate points.…”

“…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). 4) The study of conjugate points along abnormal geodesics is completely open.…”

“…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…”

“…Theorem 23 (Warner-regularity of the sub-Riemannian exponential map [BK21]). Let M be a sub-Riemannian manifold and p ∈ M. Then, (R1) The map exp p is C ∞ on U p = Dom(exp p ) and, for all λ 0 ∈ U p \ H −1 p (0) and all t ∈ I p,λ 0 , we have d tλ 0 exp p (ṙ p,λ 0 (t)) = 0 exp p (tλ 0 ) ; (R2) The map…”