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Multi-linear Formulation of Differential Geometry and Matris Regularizations
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.ORCID iD: 0000-0002-8727-2169
Sogang University, South Korea .
Max Planck Institute for Gravitational Physics, Germany .
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 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 Weingartens 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.

Place, publisher, year, edition, pages
International Press , 2012. Vol. 91, no 1, 1-39 p.
National Category
Natural Sciences
URN: urn:nbn:se:liu:diva-81838ISI: 000308046900001OAI: diva2:556585
Available from: 2012-09-25 Created: 2012-09-24 Last updated: 2013-08-29

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Arnlind, Joakim
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