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Zbl 1121.35111
Constantin, A.; Kappeler, T.; Kolev, B.; Topalov, P.
On geodesic exponential maps of the Virasoro group. (English)
[J] Ann. Global Anal. Geom. 31, No. 2, 155-180 (2007). ISSN 0232-704X; ISSN 1572-9060/e

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 2000:: *35Q35 Other equations arising in fluid mechanics
37K30 Relations with algebraic structures
37K25 Relations with differential geometry
58B25 Group structures and generalizations on infinite-dim. manifolds

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

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