The search for non-smooth geodesics in subriemannian geometry

Abstract

In Euclidean geometry length-minimizing curves are straight lines. In Riemannian geometry, length-minimizers need no longer be lines in the Euclidean sense, as the Riemannian manifold itself may be curved. Nonetheless length-minimizers in the Riemannian case are still smooth curves. In the setting of subriemannian geometry the Riemannian argument for smoothness of length-minimizers fails. No other proof of smoothness has been found either. However, there are also no known examples of non-smooth subriemannian length-minimizers. I will be discussing some of the basic concepts related to studying curves and length in subriemannian geometry. In particular, I will describe some restrictions on the class of curves that could potentially be examples of non-smooth length-minimizers.