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Multi-linear Formulation of Differential Geometry and Matris Regularizations
Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska högskolan.ORCID-id: 0000-0002-8727-2169
Sogang University, South Korea .
Max Planck Institute for Gravitational Physics, Germany .
2012 (engelsk)Inngår i: Journal of differential geometry, ISSN 0022-040X, E-ISSN 1945-743X, Vol. 91, nr 1, s. 1-39Artikkel i tidsskrift (Fagfellevurdert) 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.

sted, utgiver, år, opplag, sider
International Press , 2012. Vol. 91, nr 1, s. 1-39
HSV kategori
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URN: urn:nbn:se:liu:diva-81838ISI: 000308046900001OAI: oai:DiVA.org:liu-81838DiVA, id: diva2:556585
Tilgjengelig fra: 2012-09-25 Laget: 2012-09-24 Sist oppdatert: 2017-12-07

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