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Astérisque - Titles - 390 (2017) 277-301

Titles < 2017 < 390

Séminaire Bourbaki, volume 2015/2016, exposés 1104-1119
Astérisque 390 (2017), xi+533 pages
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Presentation, Summary

Exposé 1113 : Singulières minimisantes en géométrie sous-riemannienne
Ludovic RIFFORD
Astérisque 390 (2017), 277-301
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Résumé :
L'un des problèmes fondamentaux en géométrie sous-riemannienne porte sur la régularité des géodésiques minimisantes. Une structure sous-riemannienne sur une variété correspond à la donnée d'une distribution totalement non holonome et d'une métrique sur celle-ci. La propriété de non-intégrabilité de la distribution garantit l'existence de courbes horizontales, c'est-à-dire tangentes à la distribution, entre tous points et la métrique permet de définir une notion de distance sur la variété. Comme en géométrie riemannienne, sous des hypothèses appropriées on peut montrer l'existence de courbes horizontales minimisant la longueur, mais contrairement au cas riemannien de telles courbes ne sont pas nécessairement solutions d'une équation géodésique. Ce phénomène est à l'origine du problème de régularité des singulières minimisantes en géométrie sous-riemannienne.

Mots-clefs : Géométrie sous-riemannienne, courbes horizontale singulières, géodésiques singulières minimisantes, conjecture de Sard.

Abstract:
Exposé 1113 : Singular minimizers in sub-Riemannian geometry
One of the major open problems in sub-Riemannian geometry is the regularity of minimizing curves. A sub-Riemannian structure on some manifold is given by a totally nonholonomic distribution equiped with a metric. The property of total nonholonomy guarantees the existence of horizontal curves, that is curves which are tangent to the distribution, between any pair of points and the metric allows to define a distance. Like in the Riemannian setting, under appropriate assumptions, we can show the existence of a minimizing horizontal curve between two points but in the sub-Riemannian setting such minimizers are not necessarily solutions to some ``geodesic equation''. This phenomenon raises the issue of the regularity of singular minimizers.

Keywords: Sub-Riemannian geometry, singular horizontal curves, singular minimizing geodesics, Sard conjecture.

Class. math. : 53C17 49K21 28A15.


ISSN : 0303-1179
Publié avec le concours de : Centre National de la Recherche Scientifique

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