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Multi linear formulation of differential geometry and matrix regularizations

MPG-Autoren
http://pubman.mpdl.mpg.de/cone/persons/resource/persons4309

Arnlind,  Joakim
Quantum Gravity & Unified Theories, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons20689

Huisken,  Gerhard
Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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Volltexte (frei zugänglich)

1009.4779
(Preprint), 384KB

JDG91_1.pdf
(beliebiger Volltext), 298KB

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Zitation

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.


Zitierlink: http://hdl.handle.net/11858/00-001M-0000-000E-EAFB-C
Zusammenfassung
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.