Sub-Riemannian manifolds and their abnormal curves

Abstract

One of the features distinguishing sub-Riemannian geometry from Riemannian geometry is the existence of so called abnormal curves, which are Lipschitz curves with the property that first order variations are constrained to some lower dimensional space. The abnormal curves are the focus of some important open problems in sub-Riemannian geometry such as the regularity of length-minimizing curves and the Sard problem. In this talk I will give a brief overview of these problems and present some results on how complicated the abnormal curves can be.