Journal of Lie Theory
Vol. 8, No. 2, pp. 293-309 (1998)

On the geometry of the Virasoro-Bott group

P. W. Michor and T. S. Ratiu

PWMichor: Institut für Mathematik
Universität Wien
Strudlhofgasse 4
A-1090 Wien, Austria
and, concurrently,
Erwin Schrödinger International Institute of Mathematical Physics
Pasteurgasse 6/7
A-1090 Wien, Austria

TSRatiu: Department of Mathematics
University of California
Santa Cruz, CA 95064, USA

Abstract: 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.

Keywords: diffeomorphism group, connection, Jacobi field, symplectic structure, KdV equation

Classification (MSC91): 58D05, 58F07; 35Q53

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