Discrete curvature and the Gauss-Bonnet theorem

Arnlind, Joakim; Hoppe, Jens; Huisken, Gerhard

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Discrete curvature and the Gauss-Bonnet theorem

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Arnlind, Joakim
Quantum Gravity & Unified Theories, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

Hoppe, Jens
Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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Huisken, Gerhard
Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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1001.2223
(Preprint), 175KB

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Arnlind, J., Hoppe, J., & Huisken, G. (in preparation). Discrete curvature and the Gauss-Bonnet theorem.

Cite as: https://hdl.handle.net/11858/00-001M-0000-0012-B7F3-C

Abstract

For matrix analogues of embedded surfaces we define discrete curvatures and Euler characteristics, and a non-commutative Gauss--Bonnet theorem is shown to follow. We derive simple expressions for the discrete Gauss curvature in terms of matrices representing the embedding coordinates, and provide a large class of explicit examples illustrating the new notions.