A short proof of a Theorem by Hopf

TL;DR

This paper presents a concise proof of Hopf's theorem on the degree of the Gauss map for hypersurfaces in Euclidean space, utilizing the Chern-Gauss-Bonnet theorem to simplify the argument.

Contribution

It offers a new, streamlined proof of Hopf's theorem based on differential geometry and topological invariants, avoiding more complex traditional methods.

Findings

Proof confirms the degree of the Gauss map equals the Euler characteristic for hypersurfaces.

Utilizes the Chern-Gauss-Bonnet theorem to connect curvature and topology.

Simplifies understanding of classical differential geometry results.

Abstract

A proof based on the Chern-Gauss-Bonnet Theorem is given to Hopf Theorem concerning the degree of the Gauss map of a hypersurface in $\mathbb{R}^n$.