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A shallow water equation as a geodesic flow on the Bott-Virasoro group. (English) Zbl 0901.58022
The author proves that the Camassa-Holm equation gives rise to a geodesic flow of a certain right invariant metric on the Bott-Virasoro group. It is explained that the sectional curvature of this metric is taking positive and negative signs. In addition to these results the paper contains some rather interesting remarks.

37J35Completely integrable systems, topological structure of phase space, integration methods
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
35Q53KdV-like (Korteweg-de Vries) equations
Full Text: DOI
[1] Arnol’d, V. I.: Mathematical methods of classical mechanics. (1989)
[2] Bott, R.: On the characteristic classes of groups of diffeomorphisms. Enseign. math. 23 (1977) · Zbl 0367.57004
[3] Camassa, R.; Holm, D.: An integrable shallow water equation with peaked solutions. Phys. rev. Lett. 71 (1993) · Zbl 0972.35521
[4] Camassa, R.; Holm, D.; Hyman, J.: A new integrable shallow water equation. Adv. appl. Mech. 31 (1994) · Zbl 0808.76011
[5] Cheeger, J.; Ebin, D. G.: Comparison theorems in Riemannian geometry. (1975) · Zbl 0309.53035
[6] Gelfand, I. M.; Fuchs, D. B.: The cohomology of the Lie algebra of vector fields on a circle. Funktsional anal. Prilozhen. 2 (1968)
[7] Marsden, J.; Ratiu, T.: Introduction to mechanics and symmetry. (1995)
[8] Misiołek, G.: Stability of flows of ideal fluids and the geometry of the group of diffeomorphisms. Indiana univ. Math. J. 42 (1993) · Zbl 0799.58019
[9] G. Misiołek, Conjugate points in the Bott-Virasoro group and the KdV equation, Proc. AMS, to appear. · Zbl 06489372
[10] Ovsienko, V.; Khesin, B.: Korteweg-de Vries superequations as an Euler equation. Functional anal. Appl. 21 (1987) · Zbl 0655.58018
[11] Pressley, A.; Segal, G.: Loop groups. (1986) · Zbl 0618.22011