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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:
35Q35PDEs in connection with fluid mechanics
37K30Relations of infinite-dimensional systems with algebraic structures
37K25Relations of infinite-dimensional systems with differential geometry
58B25Group structures and generalizations on infinite-dimensional manifolds
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