We study the geodesic exponential maps corresponding to Sobolev type right-invariant (weak) Riemannian metricsμ(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 Fréchet chart of the unital element e ∈Vir. In particular, the geodesic exponential map corresponding to the Korteweg-de Vries (KdV) equation (k=0) is not a local diffeomorphism near the origi
