Calculus of Variations and Geometric Measure Theory

S. Borza - W. Klingenberg

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

created by borza1 on 22 Aug 2024

[BibTeX]

Published Paper

Inserted: 22 aug 2024
Last Updated: 22 aug 2024

Journal: Nonlinear Analysis
Volume: 239
Pages: 113421
Year: 2024
Doi: https://doi.org/10.1016/j.na.2023.113421

ArXiv: 2206.05586 PDF

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.