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#### Multi linear formulation of differential geometry and matrix regularizations

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1009.4779

(Preprint), 384KB

JDG91_1.pdf

(Any fulltext), 298KB

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##### Citation

Arnlind, J., Hoppe, J., & Huisken, G. (2012). Multi linear formulation of differential
geometry and matrix regularizations.* Journal of differential geometry,* *91*(1
), 1-39. Retrieved from http://projecteuclid.org/euclid.jdg/1343133699.

Cite as: https://hdl.handle.net/11858/00-001M-0000-000E-EAFB-C

##### 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 a large class of explicit examples is 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.