Riemannian geometry of ${\rm Diff}(S^1)/S^1$ and representations of the   Virasoro algebra
  The main result of the paper is a computation of the Ricci curvature of $\DS/S^1$. Unlike earlier results on the subject, we do not use the K\"{a}hler structure symmetries to compute the Ricci curvature, but rather rely on classical finite-dimensional results of Nomizu et al on Riemannian geometry of homogeneous spaces. 
M.Gordina, P.Lescot