Commentarii Mathematici Helvetici
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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, UNITED KINGDOM
(2) Centre de Mathématiques et Informatique, Université de Provence, 39, rue Joliot-Curie, F-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