Connected components of the space of proper gradient vector fields

TL;DR

This paper demonstrates that in the space of proper gradient vector fields on Euclidean space, certain fields are homotopic as proper maps but not as proper gradient maps, revealing a nuanced difference in their topological classifications.

Contribution

It introduces a distinction between homotopy classes of proper maps and proper gradient maps, showing they are not equivalent in this context.

Findings

Existence of two proper gradient vector fields homotopic as proper maps but not as proper gradient maps.

Highlights a difference in homotopy classifications between proper maps and proper gradient maps.

Provides insight into the topology of gradient vector fields on Euclidean spaces.

Abstract

We show that there exist two proper gradient vector fields on $\mathbb{R}^n$ which are homotopic in the category of proper maps but not homotopic in the category of proper gradient maps.