Variational aspects of the geodesics problem in sub-Riemannian geometry
✍ Scribed by Paolo Piccione; Daniel V. Tausk
- Publisher
- Elsevier Science
- Year
- 2001
- Tongue
- English
- Weight
- 186 KB
- Volume
- 39
- Category
- Article
- ISSN
- 0393-0440
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✦ Synopsis
We study the local geometry of the space of horizontal curves with endpoints freely varying in two given submanifolds P and Q of a manifold M endowed with a distribution D ⊂ 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, Progress in Mathematics, Birkhauser, Boston, 1984, Theorem 1.17] 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 [Memoirs Am. Math. Soc. 564 (1995) 118] that shows the global distance minimizing property of sufficiently small portions of normal extremizers between a point and a submanifold.