Multi linear formulation of differential geometry and matrix regularizations

TL;DR

This paper reformulates key concepts of differential geometry for embedded manifolds using multi linear algebra, introduces matrix analogues for curvature and topology, and establishes a non-commutative Gauss-Bonnet theorem with practical examples.

Contribution

It provides a novel algebraic framework for differential geometry using multi linear algebra and extends classical geometric results to matrix analogues with explicit formulas.

Findings

Derived algebraic expressions for curvature and fundamental equations.

Defined discrete curvatures and Euler characteristics for matrix analogues.

Proved a non-commutative Gauss-Bonnet theorem and related eigenvalue bounds.

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.