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

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.

arXiv comment19 pages, v2: acknowledgements updated, v3: 22 pages, major improvements, title changed to a more informative one, hypothesis of Lemma 31. fixed, introduction and last section significantly expanded, Remark 28. added, typos corrected