Element of sub-riemannian geometry and its applications

Abstract

The goal of this work is to explain the elements of sub-Riemannian geometry and present some of its applications. We start with the concept of distribution, then explain the Frobenius theorem. After that, we focus on sub-Riemannian structures and prove the Chow theorem, central in the study of connectivity through horizontal geodesics. In the last part of the thesis, we consider the $ G $-principal bundles endowed with compatible sub-Riemannian structures, prove the theorem on normal geodesics of bundle type sub-Riemannian metrics, and study Wong's equations.