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

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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.## References

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## Additional Information

**G. Marí Beffa**- Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
- Email: maribeff@math.wisc.edu
- 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
- © Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**134**(2006), 779-791 - MSC (2000): Primary 37K25; Secondary 37K05, 37K10, 53A55
- DOI: https://doi.org/10.1090/S0002-9939-05-07998-0
- MathSciNet review: 2180896