Existence of Horizontal Immersions in Fat Distributions

TL;DR

This paper introduces a numerical invariant called degree for corank 2 fat distributions and proves the existence of horizontal immersions into degree 2 fat distributions and quaternionic contact structures using Gromov's h-principle.

Contribution

It defines the degree invariant for fat distributions and establishes the existence of horizontal immersions into these structures, expanding understanding of their geometric properties.

Findings

Existence of horizontal immersions into degree 2 fat distributions.

Introduction of the degree invariant for corank 2 fat distributions.

Application of Gromov's h-principle techniques to contact and quaternionic structures.

Abstract

Contact structures, as well as their holomorphic and quaternionic counterparts are the primary examples of strongly bracket generating (or fat) distributions. In this article we associate a numerical invariant to corank $2$ fat distribution on manifolds, referred to as \emph{degree} of the distribution. The real distribution underlying a holomorphic contact structure is of degree $2$. Using Gromov's sheaf theoretic and analytic techniques of $h$-principle, we prove the existence of horizontal immersions of an arbitrary manifold into degree $2$ fat distributions and the quaternionic contact structures. We also study immersions of a contact manifold inducing the given contact structure.