The $h$-Principle for Maps Transverse to Bracket-Generating Distributions

TL;DR

This paper proves that maps transverse to bracket-generating distributions on manifolds satisfy the complete h-principle, advancing understanding in geometric topology and differential geometry.

Contribution

It establishes the complete h-principle for maps transverse to bracket-generating distributions, partially answering Gromov's longstanding question.

Findings

Maps transverse to bracket-generating distributions satisfy the h-principle.

The result applies to manifolds with constant growth distributions.

Partially settles Gromov's question on the h-principle for such maps.

Abstract

Given a smooth bracket-generating distribution $\mathcal{D}$ of constant growth on a manifold $M$, we prove that maps from an arbitrary manifold $\Sigma$ to $M$, which are transverse to $\mathcal{D}$, satisfy the complete $h$-principle. This partially settles a question posed by M. Gromov.