Existence of horizontal immersions in fat distributions

Abstract

Contact structures, as well as their holomorphic and quaternionic counterparts, are the primary examples of strongly bracket generating (or fat) distributions. In this paper, we associate a numerical invariant to corank-[Formula: see text] fat distribution on manifolds, referred to as the degree of the distribution. The real distribution underlying a holomorphic contact structure is of degree [Formula: see text]. Using Gromov’s sheaf theoretic and analytic techniques of [Formula: see text]-principle, we prove the existence of horizontal immersions of an arbitrary manifold into degree [Formula: see text] fat distributions and the quaternionic contact structures. We also study isocontact immersions, i.e. immersions that pull back a fat distribution to a given contact structure.