Shortest paths for sub-Riemannian metrics on rank-two distributions 论文

摘要

We study length-minimizing arcs in sub-Riemannian manifolds (M;E;G) whose metric G is defined on a rank-two bracket-generating distribution E. It is well known that all length-minimizing arcs are extremals, and that these extremals are either "normal" or "abnormal." Normal extremals are locally optimal, in the sense that every sufficiently short piece of such an extremal is a minimizer. The question whether every length-minimizer is a normal extremal remained open for several years, and was recently settled by R. Montgomery, who exhibited a counterexample. But Montgomery's geometric optimality proof depends heavily on special properties of his example and still leaves open the question whether abnormal minimizers are an exceptional phenomenon or a common occurrence. We present an analytic technique for proving local optimality of a large class of abnormal extremals that we call "regular." Our technique is based on (a) a "normal form theorem," stating that, locally, a regular abnormal e...