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

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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https://hal.archives-ouvertes.fr/hal-01961959
Contributor : Ludovic Sacchelli <>
Submitted on : Monday, April 29, 2019 - 11:27:56 AM
Last modification on : Wednesday, July 24, 2019 - 10:50:42 PM

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  • HAL Id : hal-01961959, version 3
  • ARXIV : 1812.11340

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Ludovic Sacchelli. Short geodesics losing optimality in contact sub-Riemannian manifolds and stability of the 5-dimensional caustic. SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2019, 57 (4), pp.2362-2391. ⟨hal-01961959v3⟩

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