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

Ludovic Sacchelli (1)

1 LIS - Laboratoire d'Informatique et des Systèmes (LIS) (Marseille, Toulon)

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.

Mots clés: Sub-Riemannian geometry; Conjugate locus; Contact distributions

Domaines: Géométrie différentielle [math.DG]; Optimisation et contrôle [math.OC]

Ludovic Sacchelli. Short geodesics losing optimality in contact sub-Riemannian manifolds and stability of the 5-dimensional caustic. 2018. ⟨hal-01961959v1⟩