  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.
