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Multi linear formulation of differential geometry and matrix regularizations
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Max Planck Institute for Gravitational Physics.
2012 (English)In: Journal of differential geometry, ISSN 0022-040X, E-ISSN 1945-743X, Vol. 91, no 1, 1-39 p.Article in journal (Refereed) Published
Abstract [en]

We prove that many aspects of the differential geometry of em-bedded Riemannian manifolds can be formulated in terms of multilinear algebraic structures on the space of smooth functions. Inparticular, we find algebraic expressions for Weingarten’s formula,the Ricci curvature and the Codazzi-Mainardi equations.For matrix analogues of embedded surfaces we define discretecurvatures and Euler characteristics, and a non-commutative Gauss–Bonnet theorem is shown to follow. We derive simple expressionsfor the discrete Gauss curvature in terms of matrices representingthe embedding coordinates, and explicit examples are provided.Furthermore, we illustrate the fact that techniques from differen-tial geometry can carry over to matrix analogues by proving thata bound on the discrete Gauss curvature implies a bound on theeigenvalues of the discrete Laplace operator.

Place, publisher, year, edition, pages
2012. Vol. 91, no 1, 1-39 p.
National Category
URN: urn:nbn:se:kth:diva-84595ISI: 000308046900001OAI: diva2:499396

QC 20130204

Available from: 2012-02-15 Created: 2012-02-13 Last updated: 2013-02-04Bibliographically approved

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