Shortest Paths for Sub-Riemannian Metrics on Rank-Two Distributions

This work studies length-minimizing arcs in sub-Riemannian manifolds $(M, E, G)$ where the metric $G$ is defined on a rank-two bracket-generating distribution $E$. The authors define a large class of abnormal extremals---the "regular" abnormal extremals---and present an analytic technique for proving their local optimality. If $E$ satisfies a mild additional restriction-valid in particular for all regular two-dimensional distributions and for generic two-dimensional distributions---then regular abnormal extremals are "typical," in a sense made precise in the text. So the optimality result implies that the abnormal minimizers are ubiquitous rather than exceptional.