Short geodesics losing optimality in contact sub-Riemannian manifolds and stability of the 5-dimensional caustic
Résumé
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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Caustics_HAL_V3.pdf (3.7 Mo)
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A4_1.pdf (741.75 Ko)
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D4plus_1.pdf (764.16 Ko)
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D4plus_2.pdf (1.65 Mo)
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Phi_anorm.pdf (3.98 Ko)
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Phi_norm.pdf (4.74 Ko)
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Origine : Fichiers produits par l'(les) auteur(s)
Origine : Fichiers produits par l'(les) auteur(s)