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

TL;DR

This paper demonstrates that the sub-Riemannian exponential map is locally non-injective near certain critical points, providing a new characterization of conjugate points in ideal sub-Riemannian manifolds through metric structure analysis.

Contribution

It establishes the non-injectivity of the exponential map near critical points and characterizes conjugate points via the metric structure in sub-Riemannian geometry.

Findings

Exponential map is not injective near certain critical points.

Conjugate points are characterized by metric structure.

Uses Hilbert invariant integral in the proof.

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.