Variational aspects of the geodesic problem in sub-Riemannian geometry

Details Variational aspects of the geodesic problem in sub-Riemannian geometry Variational aspects of the geodesic problem in sub-Riemannian geometry

1999-11-26

arxiv.org/abs/math/9911215v3

Mathematics

Differential Geometry

LaTeX2e, amsart class, 25 pages Replacement on Jan 5th 2000: added Appendix B Replacement on May 23rd 2000: modified Abstract

Scientific paper

We study the local geometry of the space of horizontal curves with endpoints freely varying in two given submanifolds $\mathcal P$ and $\mathcal Q$ of a manifold $\mathcal M$ endowed with a distribution $\mathcal D\subset T\M$. We give a different proof, that holds in a more general context, of a result by Bismut (Large Deviations and the Malliavin Calculus, Birkhauser, 1984) stating that the normal extremizers that are not abnormal are critical points of the sub-Riemannian action functional. We use the Lagrangian multipliers method in a Hilbert manifold setting, which leads to a characterization of the abnormal extremizers (critical points of the endpoint map) as curves where the linear constraint fails to be regular. Finally, we describe a modification of a result by Liu and Sussmann that shows the global distance minimizing property of sufficiently small portions of normal extremizers between a point and a submanifold.

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