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Zbl 1037.37032
Constantin, Adrian; Kolev, Boris
Geodesic flow on the diffeomorphism group of the circle. (English)
[J] Comment. Math. Helv. 78, No. 4, 787-804 (2003). ISSN 0010-2571; ISSN 1420-8946/e

Summary: We show that certain right-invariant metrics endow the infinite-dimensional Lie group of all smooth orientation-preserving diffeomorphisms of the circle with a Riemannian structure. The study of the Riemannian exponential map allows us to prove infinite-dimensional counterparts of results from classical Riemannian geometry: The Riemannian exponential map is a smooth local diffeomorphism and the length-minimizing property of the geodesics holds.

MSC 2000:: *37K65 Hamiltonian systems on groups of diffeomorphisms etc.
35Q35 Other equations arising in fluid mechanics
58B25 Group structures and generalizations on infinite-dim. manifolds
53D25 Geodesic flows

Keywords: geodesic flow; diffeomorphism group of the circle; inviscid Burgers equation; Camassa-Holm equation

Cited in: Zbl 1186.35003

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