Geodesics in subriemannian geometry

Abstract

This work is a short, self-contained introduction to subriemannian geometry with special emphasis on Chow's Theorem.<br />At the beginning, the definitions of a subriemannian geometry, horizontal vectorfields and horizontal curves are given. Then the question arises:<br />Can any two points be connected by a horizontal curve? Chow's Theorem gives an affirmative answer for bracket generating distributions. (A distribution is called bracket generating if horizontal vectorfields and their iterated Lie brackets span the whole tangent space.) We present three different proofs of Chow's Theorem; each one is interesting in its own. The first proof is based on the theory of Stefan and Sussmann regarding integrability of singular distributions. The second proof is elementary and gives some insight in the shape of subriemannian balls. The third proof is based on infinite dimensional analysis of the endpoint map.<br />Finally I point out one of the main differences between riemannian geometry and subriemannian geometry, the existence of geodesics which are not solutions to Hamilton´s differential equations. Indeed till 1990 there have been a lot of false theorems asserting that every subriemannian geodesic is a solution to Hamilton´s differential equation in the literature, for example from Rayner (1967), Taylor (1989) and Strichartz (1986,1989). The first counterexample, which I will present in the begin of the second part was given by Montgomery in 1994. Later Liu and Sussmann gave a much simpler proof for a more general result.<br />In the end of my work I will show that for a special class the existence of this abnormal geodesics can be ruled out.