On geodesic exponential maps of the Virasoro group

We study the geodesic exponential maps corresponding to Sobolev type right-invariant (weak) Riemannian metrics μ^(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 origin. A. Constantin: Supported in part by the European Community through the FP6 Marie Curie RTN ENIGMA (MRTN-CT-2004-5652). T. Kappeler: Supported in part by the SNSF, the programme SPECT, and the European Community through the FP6 Marie Curie RTN ENIGMA (MRTN-CT-2004-5652)