Variational aspects of the geodesics problem in sub-Riemannian geometry

We study the local geometry of the space of horizontal curves with endpoint s freely varying in two given submanifolds P and Q of a manifold M endowed with a distribution D subset of TM. We give a different proof, that holds i n a more general context, of a result by Bismut [Large Deviations and the M alliavin Calculus, Progress in Mathematics, Birkhauser, Boston, 1984, Theor em 1.17] stating that the normal extremizers that are not abnormal are crit ical points of the sub-Riemannian action functional. We use the Lagrangian multipliers method in a Hilbert manifold setting, which leads to a characte rization of the abnormal extremizers (critical points of the endpoint map) as curves where the linear constraint fails to be regular. Finally, we desc ribe a modification of a result by Liu and Sussmann [Memoirs Am. Math. Soc. 564 (1995) 118] that shows the global distance minimizing property of suff iciently small portions of normal extremizers between a point and a submani fold. (C) 2001 Elsevier Science B.V. All rights reserved.