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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Poisson brackets associated to the conformal geometry of curves

Author: G. Marí Beffa
Journal: Trans. Amer. Math. Soc. 357 (2005), 2799-2827
MSC (2000): Primary 37K25, 37K05, 37K10; Secondary 53A55
Published electronically: September 23, 2004
MathSciNet review: 2139528
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Abstract: In this paper we present an invariant moving frame, in the group theoretical sense, along curves in the Möbius sphere. This moving frame will describe the relationship between all conformal differential invariants for curves that appear in the literature. Using this frame we first show that the Kac-Moody Poisson bracket on $Lo(n+1,1)^\ast$ can be Poisson reduced to the space of conformal differential invariants of curves. The resulting bracket will be the conformal analogue of the Adler-Gel'fand-Dikii bracket. Secondly, a conformally invariant flow of curves induces naturally an evolution on the differential invariants of the flow. We give the conditions on the invariant flow ensuring that the induced evolution is Hamiltonian with respect to the reduced Poisson bracket. Because of a certain parallelism with the Euclidean case we study what we call Frenet and natural cases. We comment on the implications for completely integrable systems, and describe conformal analogues of the Hasimoto transformation.

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

Keywords: Invariant evolutions of curves, conformal curves, Poisson brackets, differential invariants, completely integrable PDEs
Received by editor(s): June 11, 2003
Received by editor(s) in revised form: November 14, 2003
Published electronically: September 23, 2004
Article copyright: © Copyright 2004 American Mathematical Society