On geodesic exponential maps of the Virasoro group

We study the geodesic exponential maps corresponding to Sobolev type right-invariant (weak) Riemannian metrics mu((k)) (k >= 0) on the Virasoro group Vir and show that for k >= 2, but not for k = 0, 1, each of them defines a smooth Frechet chart of the unital element e is an element of 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.