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Constantin, A.; Kappeler, T.; Kolev, B.; Topalov, P.

On geodesic exponential maps of the Virasoro group. (English) Zbl 1121.35111

Ann. Global Anal. Geom. 31, No. 2, 155-180 (2007).
Summary: We study the geodesic exponential maps corresponding to Sobolev type right-invariant (weak) Riemannian metrics \(\mu^{(k)}\) \((k \geq\) 0) on the Virasoro group \(Vir\) and show that for \(k \geq 2\), but not for \(k = 0,1\), each of them defines a smooth Fréchet chart of the unital element \(e \in\) Vir. In particular, the geodesic exponential map corresponding to the Korteweg-de Vries (KdV) equation \((k = 0)\) is not a local diffeomorphism near the origin.

Cited in 110 Documents

MSC:

35Q35 PDEs in connection with fluid mechanics
37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
58B25 Group structures and generalizations on infinite-dimensional manifolds

Keywords:

Euler equation; KdV equation; Fréchet manifold; Virasoro-Bott group; geodesic flow; Camassa-Holm equation
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References:

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