Variational Aspects of the Geodesic Problem in Sub-riemannian Geometry

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 ⊂ TM. We give a different proof, that holds in a more general context, of a result by Bismut [2, 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 [9] that shows the global distance minimizing property of sufficiently small portions of normal extremizers between a point and a submanifold.