Multi-linear formulation of differential geometry and matrix regularizations

Joakim Arnlind Link¨oping University Jens Hoppe Sogang University Gerhard Huisken Max Planck Institute for Gravitational Physics

Differential Geometry mathscidoc:1609.10263

Journal of Differential Geometry, 91, (1), 1-39, 2012
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 GaussBonnet 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.
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@inproceedings{joakim2012multi-linear,
  title={MULTI-LINEAR FORMULATION OF DIFFERENTIAL GEOMETRY AND MATRIX REGULARIZATIONS},
  author={Joakim Arnlind, Jens Hoppe, and Gerhard Huisken},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160913225340100201925},
  booktitle={Journal of Differential Geometry},
  volume={91},
  number={1},
  pages={1-39},
  year={2012},
}
Joakim Arnlind, Jens Hoppe, and Gerhard Huisken. MULTI-LINEAR FORMULATION OF DIFFERENTIAL GEOMETRY AND MATRIX REGULARIZATIONS. 2012. Vol. 91. In Journal of Differential Geometry. pp.1-39. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160913225340100201925.
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