TOTALLY NONHOLONOMIC DISTRIBUTIONS AND SUB-RIEMANNIAN STRUCTURES

Abstract

In this paper, we study totally non-holonomic distributions and sub-riemannian structure on Rn (n = 1, 2, 3, 4, ...) and on the manifolds of not null curvature such as Tn and Sn with n more or equal than 3, in order to look at this structure on a manifold like a restricting of a Riemannian metric on a totally non-holonomic distribution of this manifold. On Rn, it always possible to construct this structure thanks to her multitude of smooth distributions and vector elds. On the manifolds of not null curvature, we de ne this structure on the kernel of a non-degenerate 1-form or of contact form on this manifold.