Double Forms, Curvature Integrals and the Gauss-Bonnet Formula

TL;DR

This paper revisits the Gauss-Bonnet Formula for Riemannian manifolds using double forms, clarifying the boundary term's geometric nature and providing new examples and applications.

Contribution

It introduces a double forms formalism to analyze the Gauss-Bonnet Formula, offering new insights into the boundary term and its geometric interpretation.

Findings

Clarification of the boundary term in the Gauss-Bonnet Formula

Application of double forms to higher-dimensional manifolds

Examples illustrating the geometric nature of curvature integrals

Abstract

The Gauss-Bonnet Formula is a significant achievement in 19th century differential geometry for the case of surfaces and the 20th century cumulative work of H. Hopf, W. Fenchel, C. B. Allendoerfer, A. Weil and S.S. Chern for higher-dimensional Riemannian manifolds. It relates the Euler characteristic of a Riemannian manifold to a curvature integral over the manifold plus a somewhat enigmatic boundary term. In this paper, we revisit the formula using the formalism of double forms, a tool introduced by de Rham, and further developed by Kulkarni, Thorpe, and Gray. We explore the geometric nature of the boundary term and provide some examples and applications.