Double Forms, Curvature Integrals and the Gauss–Bonnet Formula

Marc Troyanov ()
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Marc Troyanov: Institut de Mathématiques EPFL

Chapter Chapter 4 in Surveys in Geometry II, 2024, pp 93-143 from Springer

Abstract: Abstract The Gauss–Bonnet Formula is a significant achievement in nineteenth century differential geometry for the case of surfaces and the twentieth 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 chapter, 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.

Keywords: Gauss–Bonnet formula; Double forms; Curvature integrals; 58A10; 53C20 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/978-3-031-43510-2_4

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