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  Multi linear formulation of differential geometry and matrix regularizations

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.

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Item Permalink: http://hdl.handle.net/11858/00-001M-0000-000E-EAFB-C Version Permalink: http://hdl.handle.net/11858/00-001M-0000-000E-EAFC-A
Genre: Journal Article

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1009.4779 (Preprint), 384KB
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 Creators:
Arnlind, Joakim1, Author              
Hoppe, Jens, Author
Huisken, Gerhard2, Author              
Affiliations:
1Quantum Gravity & Unified Theories, AEI-Golm, MPI for Gravitational Physics, Max Planck Society, escidoc:24014              
2Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society, escidoc:24012              

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Free keywords: Mathematics, Differential Geometry, math.DG,High Energy Physics - Theory, hep-th,
 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.

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 Dates: 2010-09-242012
 Publication Status: Published in print
 Pages: -
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 Rev. Method: -
 Identifiers: arXiv: 1009.4779
URI: http://projecteuclid.org/euclid.jdg/1343133699
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Title: Journal of differential geometry
Source Genre: Journal
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Publ. Info: Bethlehem, Pa. : Lehigh University
Pages: - Volume / Issue: 91 (1 ) Sequence Number: - Start / End Page: 1 - 39 Identifier: ISSN: 0022-040X
CoNE: http://pubman.mpdl.mpg.de/cone/journals/resource/954925412864