Regularity and Continuity properties of the sub-Riemannian exponential map

TL;DR

This paper investigates the regularity and continuity of the sub-Riemannian exponential map, demonstrating key properties and implications for the Heisenberg group, with implications for geometric analysis.

Contribution

It establishes regularity and continuity properties of the sub-Riemannian exponential map using Jacobi fields and Maslov index, and shows non-injectivity near conjugate vectors in the Heisenberg group.

Findings

Proves regularity and continuity properties of the sub-Riemannian exponential map.

Shows the exponential map of the Heisenberg group is not injective near conjugate vectors.

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.