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 $\text{Emb}(\mathbb{R},\mathbb{R})$ 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 $\text{Diff}(\mathbb{R})$, $\text{Diff}(S^1)$, and the Virasoro-Bott group. Many of these results are well known, the emphasis is on conciseness and clarity.