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Poisson geometry of differential invariants of curves in some nonsemisimple homogeneous spaces

Author: G. Marí Beffa
Journal: Proc. Amer. Math. Soc. 134 (2006), 779-791
MSC (2000): Primary 37K25; Secondary 37K05, 37K10, 53A55
Published electronically: July 19, 2005
MathSciNet review: 2180896
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Abstract: In this paper we describe a family of compatible Poisson structures defined on the space of coframes (or differential invariants) of curves in flat homogeneous spaces of the form $\mathcal{M} \cong (G\ltimes\mathbb{R} ^n)/G$ where $G\subset {\mathrm{GL}}(n,\mathbb{R} )$ is semisimple. This includes Euclidean, affine, special affine, Lorentz, and symplectic geometries. We also give conditions on geometric evolutions of curves in the manifold $\mathcal{M}$ so that the induced evolution on their differential invariants is Hamiltonian with respect to our main Hamiltonian bracket.

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Additional Information

G. Marí Beffa
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

Keywords: Invariant evolutions of curves, homogeneous spaces, infinite dimensional Poisson geometry, differential invariants, completely integrable PDEs
Received by editor(s): August 20, 2004
Received by editor(s) in revised form: October 15, 2004
Published electronically: July 19, 2005
Communicated by: Jozef Dodziuk
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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