On the classical geometry of embedded surfaces in terms of Poisson brackets

Arnlind, Joakim; Hoppe, Jens; Huisken, Gerhard

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On the classical geometry of embedded surfaces in terms of Poisson brackets

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

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

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

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1001.1604
(Preprint), 145KB

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Arnlind, J., Hoppe, J., & Huisken, G. (in preparation). On the classical geometry of embedded surfaces in terms of Poisson brackets.

Cite as: https://hdl.handle.net/11858/00-001M-0000-0012-BD34-6

Abstract

We consider surfaces embedded in a Riemannian manifold of arbitrary dimension and prove that many aspects of their differential geometry can be expressed in terms of a Poisson algebraic structure on the space of smooth functions of the surface. In particular, we find algebraic formulas for Weingarten's equations, the complex structure and the Gaussian curvature.