Partial immersions and partially free maps

TL;DR

This paper explores the relationship between partial immersions and partially free maps, demonstrating how to construct the latter from the former and applying this to specific geometric structures like contact manifolds.

Contribution

It shows how to build partially free maps from partial immersions and proves their existence in critical dimensions for important distributions.

Findings

Partially free maps can be constructed from partial immersions.

Existence of partial immersions in critical dimension for contact structures.

Application to distributions in geometric analysis.

Abstract

In a recent paper~\cite{DDL10} we studied basic properties of partial immersions and partially free maps, a generalization of free maps introduced first by Gromov in~\cite{Gro70}. In this short note we show how to build partially free maps out of partial immersions and use this fact to prove that the partially free maps in critical dimension introduced in Theorems 1.1-1.3 of~\cite{DDL10} for three important types of distributions can actually be built out of partial immersions. Finally, we show that the canonical contact structure on $\bR^{2n+1}$ admits partial immersions in critical dimension for every $n$.