SHORTEST PATHS FOR SUB-RIEMANNIAN METRICS ON RANK-2 DISTRIBUTIONS - INTRODUCTION

We study length-minimizing arcs in sub-Riemannian manifolds (M, E, G) whose metric G is defined on a rank-two bracket-generating distributio n E. It is well known that all length-minimizing arcs are extrema ls, and that these extremals are either ''normal'' or ''abnormal.'' Normal extremals are locally optimal, in the sense that every sufficiently s hort piece of such an extremal is a minimizer. The question whether ev ery length-minimizer is a normal extremal remained open for several ye ars, and was recently settled by R. Montgomery, who exhibited a counte rexample. But Montgomery's geometric optimality proof depends heavily on special properties of his example and still leaves open the questio n whether abnormal minimizers are an exceptional phenomenon or a commo n occurrence. We present an analytic technique for proving local optim ality 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 extremal can always be put in a special form by a suitable change of coordinates, and (b) an inequality showin g that, once a trajectory is in this special form? then local optimali ty follows. Using this approach we prove that regular abnormal extrema ls are locally optimal. If E satisfies a mild additional restriction - valid in particular for all regular 2-dimensional distributions and f or generic 2-dimensional distributions - then regular abnormal extrema ls are ''typical'' (in a sense made precise in the text), so our resul t implies that the abnormal minimizers are ubiquitous rather than exce ptional. We also discuss some related issues, and in particular show, by means of an example, that a smooth abnormal extremal need not be lo cally optimal, even if in addition it belongs to the class - recently studied by Bryant and Hsu - of C-1-rigid curves.