Singularités en géométrie sous-riemannienne

Abstract : We investigate the relationship between features of of sub-Riemannian geometry and an array of singularities that typically arise in this context.With sub-Riemannian Whitney theorems, we ensure the existence of global extensions of horizontal curves defined on closed set by requiring a non-singularity hypothesis on the endpoint-map of the nilpotent approximation of the manifold to be satisfied.We apply perturbative methods to obtain asymptotics on the length of short locally-length-minimizing curves losing optimality in contact sub-Riemannian manifolds of arbitrary dimension. We describe the geometry of the singular set and prove its stability in the case of manifolds of dimension 5.We propose a construction to define line fields using pairs of vector fields. This provides a natural topology to study the stability of singularities of line fields on surfaces.
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Ludovic Sacchelli. Singularités en géométrie sous-riemannienne. Géométrie différentielle [math.DG]. Université Paris-Saclay, 2018. Français. ⟨NNT : 2018SACLX050⟩. ⟨tel-01893068⟩

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