Submanifolds and differential forms on Carnot manifolds, after M. Gromov and M. Rumin

Abstract

The purpose of these notes is to explain parts of Gromov's survey of Carnot-Carathedory spaces, in the light of subsequent results of M. Rumin. Among the rich material provided by Gromov, most of which pertains to analysis on metric spaces, we choose to concentrate on the H{\"o}lder equivalence problem for Carnot manifolds. The notes go to some extent into the PDE technique used by Gromov in order to construct horizontal submanifolds. Rumin's complex is explained too. The upshot is that both methods yield more or less the same conclusions as far as the H{\"o}lder equivalence problem is concerned.