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

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.

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
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