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May 2012 Multi-linear Formulation of Differential Geometry and Matris Regularizations
Joakim Arnlind, Jens Hoppe, Gerhard Huisken
J. Differential Geom. 91(1): 1-39 (May 2012). DOI: 10.4310/jdg/1343133699

Abstract

We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be formulated in terms of multi-linear algebraic structures on the space of smooth functions. In particular, we find algebraic expressions for Weingarten’s formula, the Ricci curvature, and the Codazzi-Mainardi equations.

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 explicit examples are provided. Furthermore, we illustrate the fact that techniques from differential geometry can carry over to matrix analogues by proving that a bound on the discrete Gauss curvature implies a bound on the eigenvalues of the discrete Laplace operator.

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Joakim Arnlind. Jens Hoppe. Gerhard Huisken. "Multi-linear Formulation of Differential Geometry and Matris Regularizations." J. Differential Geom. 91 (1) 1 - 39, May 2012. https://doi.org/10.4310/jdg/1343133699

Information

Published: May 2012
First available in Project Euclid: 24 July 2012

zbMATH: 1252.53024
MathSciNet: MR2944960
Digital Object Identifier: 10.4310/jdg/1343133699

Rights: Copyright © 2012 Lehigh University

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Vol.91 • No. 1 • May 2012
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