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The Euler-Poincaré Characteristic and the Gauss-Bonnet Theorem

In: An Introduction to Differential Manifolds

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  • Jacques Lafontaine

    (Université Montpellier 2, Département de Mathématiques)

Abstract

The Gauss-Bonnet theorem is at the heart of the geometry of manifolds. It mixes topology (triangulations, cohomology spaces), differential geometry (index of singular points of vector fields) and Riemannian geometry. We do not have the space to illustrate all of these ideas in detail. To keep with the spirit of the book, the proofs we give will use differential geometry to the greatest extent possible. We nonetheless believe it would be interesting to sketch a purely Riemannian proof in this introduction. The price we pay is using certain notions that have not been introduced (geodesics, geodesic curvature), of which we give the idea.

Suggested Citation

  • Jacques Lafontaine, 2015. "The Euler-Poincaré Characteristic and the Gauss-Bonnet Theorem," Springer Books, in: An Introduction to Differential Manifolds, edition 1, chapter 0, pages 323-348, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-20735-3_8
    DOI: 10.1007/978-3-319-20735-3_8
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