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.
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Dedicated to Prof. K. Nomizu on the occasion of his 70th birthday.
Research supported by DFG (Sonderforschungsbereich 288).
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Pinkall, U. Hamiltonian flows on the space of star-shaped curves. Results. Math. 27, 328–332 (1995). https://doi.org/10.1007/BF03322836
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DOI: https://doi.org/10.1007/BF03322836