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

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 conjuga...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2018-12
Main Author: Sacchelli, Ludovic
Format: Article
Language:English
Subjects:
Approximation
Conjugates
Dimensional stability
Geodesy
Lagrangian equilibrium points
Manifolds (mathematics)
Mathematical analysis
Riemann manifold
Stability analysis
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:2331-8422