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arXiv · 1812.11340

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

Ludovic Sacchelli
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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.

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Ludovic Sacchelli. 2018-12-29. Short geodesics losing optimality in contact sub-Riemannian manifolds and stability of the 5-dimensional caustic. https://arxiv.org/abs/1812.11340

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