, Volume 27, Issue 3-4, pp 328-332
Date: 24 Apr 2013

Hamiltonian flows on the space of star-shaped curves

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Abstract

We provide a geometric interpretation of the KdV equation as an evolution equation on the space of closed curves in the centroaffine plane. There is a natural symplectic structure on this space and the KdV-flow is generated by a Hamiltonian given by the total centroaffine curvature. In this way we obtain another example for a soliton equation coming naturally from a differential geometric problem [1]. Furthermore, we present a group action of the diffeomorphism group of the circle on the space of closed centroaffine curves.

Dedicated to Prof. K. Nomizu on the occasion of his 70th birthday.
Research supported by DFG (Sonderforschungsbereich 288).