**Journal of Lie Theory
**

Vol. 8, No. 2, pp. 293-309 (1998)

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On the geometry of the Virasoro-Bott group

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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

peter.michor@esi.ac.at
TSRatiu: Department of Mathematics

University of California

Santa Cruz, CA 95064, USA

ratiu@math.ucsc.edu

**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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