On the push-out space

TL;DR

This paper introduces the concept of push-out space for immersions of manifolds into Euclidean space, exploring its properties, invariance under normal holonomy, and providing geometric examples illustrating its behavior.

Contribution

It defines the push-out space for manifold immersions, studies its invariance under normal holonomy, and constructs geometric examples demonstrating its properties.

Findings

Push-out space is invariant under the normal holonomy group.

Properties of push-out space differ when the holonomy group is non-trivial.

Examples in ${ m I ext{-}R}^3$ illustrate these properties.

Abstract

Let $f:M^m\longrightarrow \Bbb R^{m+k}$ be an immersion where $M$ is a smooth connected $m$-dimensional manifold without boundary. Then we construct a subspace $\Omega(f)$ of $ \mathbb{R}^k$, namely push-out space. which corresponds to a set of embedded manifolds which are either parallel to $ f $, tubes around $ f $ or, ingeneral, partial tubes around $ f $. This space is invariant under the action of the normal holonomy group, $\mathcal{H}ol(f)$. Moreover, we construct geometrically some examples for normal holonomy group and push-out space in ${\Bbb R}^3$.These examples will show that properties of push-out space that are proved in the case $\mathcal{H}ol(f)$ is trivial, is not true in general.