PDE Real Analysis Seminar
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Date, time & place | Tuesday 10:30 - 11:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
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Organizer(s) | Yoshikazu Giga, Kazuhiro Ishige, Hiroyoshi Mitake, Tsuyoshi Yoneda |
URL | https://www.math.sci.hokudai.ac.jp/coe/sympo/pde_ra/index_en.html |
2012/04/04
10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Jens Hoppe (Sogang University / KTH Royal Institute of Technology)
Multi linear formulation of differential geometry and matrix regularizations (ENGLISH)
Jens Hoppe (Sogang University / KTH Royal Institute of Technology)
Multi linear formulation of differential geometry and matrix regularizations (ENGLISH)
[ 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.
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.