Journal of Differential Geometry

Multi-linear Formulation of Differential Geometry and Matris Regularizations

Joakim Arnlind, Jens Hoppe, and Gerhard Huisken

Full-text: Open access

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.

Article information

Source
J. Differential Geom., Volume 91, Number 1 (2012), 1-39.

Dates
First available in Project Euclid: 24 July 2012

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1343133699

Digital Object Identifier
doi:10.4310/jdg/1343133699

Mathematical Reviews number (MathSciNet)
MR2944960

Zentralblatt MATH identifier
1252.53024

Citation

Arnlind, Joakim; Hoppe, Jens; Huisken, Gerhard. Multi-linear Formulation of Differential Geometry and Matris Regularizations. J. Differential Geom. 91 (2012), no. 1, 1--39. doi:10.4310/jdg/1343133699. https://projecteuclid.org/euclid.jdg/1343133699


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