# Short geodesics losing optimality in contact sub-Riemannian manifolds and stability of the 5-dimensional caustic

## TL;DR

This paper investigates the behavior of geodesics in 5-dimensional contact sub-Riemannian manifolds, revealing unique conjugate point multiplicities and stability properties of the caustic, which differ from lower-dimensional cases.

## Contribution

It introduces a new geometric invariant and provides a detailed analysis of conjugate loci and caustic stability in higher-dimensional contact sub-Riemannian manifolds.

## Key findings

- Conjugate time can have multiplicity 2 in 5D contact manifolds.
- The metric exhibits behavior different from the classical 3D case.
- A stability analysis of the sub-Riemannian caustic is performed.

## Abstract

We study the sub-Riemannian exponential for contact distributions on manifolds of dimension greater or equal to 5. We compute an approximation of the sub-Riemannian Hamiltonian flow and show that the conjugate time can have multiplicity 2 in this case. We obtain an approximation of the first conjugate locus for small radii and introduce a geometric invariant to show that the metric for contact distributions typically exhibits an original behavior, different from the classical 3-dimensional case. We apply these methods to the case of 5-dimensional contact manifolds. We provide a stability analysis of the sub-Riemannian caustic from the Lagrangian point of view and classify the singular points of the exponential map.