Gradient versus proper gradient homotopies

We compare the sets of homotopy classes of gradient and proper gradient vector fields in the plane. Namely, we show that gradient and proper gradient homotopy classifications are essentially different. We provide a complete description of the sets of homotopy classes of gradient maps from $\mathbb{R...

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Bibliographic Details
Main Authors Bartłomiejczyk, Piotr, Nowak-Przygodzki, Piotr
Format Journal Article
LanguageEnglish
Published 28.10.2019
Subjects
Mathematics - Dynamical Systems
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Summary:We compare the sets of homotopy classes of gradient and proper gradient vector fields in the plane. Namely, we show that gradient and proper gradient homotopy classifications are essentially different. We provide a complete description of the sets of homotopy classes of gradient maps from $\mathbb{R}^n$ to $\mathbb{R}^n$ and proper gradient maps from $\mathbb{R}^2$ to $\mathbb{R}^2$ with the Brouwer degree greater or equal to zero.
DOI:10.48550/arxiv.1910.12568