Commentarii Mathematici Helvetici

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Volume 78, Issue 4, 2003, pp. 787–804
DOI: 10.1007/s00014-003-0785-6

Geodesic flow on the diffeomorphism group of the circle

Adrian Constantin[1] and Boris Kolev[2]

(1) Department of Mathematics, King's College London, Strand, WC2R 2LS, London, UK
(2) Centre de Mathématiques et Informatique, Université de Provence, 39, rue Joliot-Curie, 13453, Marseille CEDEX 13, France

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.

Keywords: Geodesic flow, diffeomorphism group of the circle

Constantin Adrian, Kolev Boris: Geodesic flow on the diffeomorphism group of the circle. Comment. Math. Helv. 78 (2003), 787-804. doi: 10.1007/s00014-003-0785-6