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Double Forms, Curvature Integrals and the Gauss–Bonnet Formula

In: Surveys in Geometry II

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  • Marc Troyanov

    (Institut de Mathématiques EPFL)

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.

Suggested Citation

  • Marc Troyanov, 2024. "Double Forms, Curvature Integrals and the Gauss–Bonnet Formula," Springer Books, in: Athanase Papadopoulos (ed.), Surveys in Geometry II, chapter 0, pages 93-143, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-43510-2_4
    DOI: 10.1007/978-3-031-43510-2_4
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