The Poincaré Lemma in Subriemannian Geometry

Abstract: This work is a short, self-contained introduction to subriemannian geometry with special emphasis on Chow's Theorem. As an application, a regularity result for the Poincar\'e Lemma is presented. At the beginning, the definitions of a subriemannian geometry, horizontal vector fields and horizontal curves are given. Then the question arises: 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 vector fields 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. Finally, the study of the endpoint map allows us to prove a regularity result for the Poincar\'e Lemma in a form suited to subriemannian geometry: If for some $r \geq 0$ all horizontal derivatives of a given function $f$ are known to be $r$ times continuously differentiable, then so is $f$. Sections 1 to 3 are the common work of Martin Bauer and Philipp Harms.