Curvature of the Virasoro-Bott group
  We consider a natural Riemannian metric on the infinite dimensional manifold of all embeddings from a manifold into a Riemannian manifold, and derive its geodesic equation in the case $\Emb(\Bbb R,\Bbb R)$ which turns out to be Burgers' equation. Then we derive the geodesic equation, the curvature, and the Jacobi equation of a right invariant Riemannian metric on an infinite dimensional Lie group, which we apply to $\Diff(\Bbb R)$, $\Diff(S^1)$, and the Virasoro-Bott group. Many of these results are well known, the emphasis is on conciseness and clarity. 
Peter W. Michor, Tudor Ratiu