Abstract
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.
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Constantin, A., Kolev, B. Geodesic flow on the diffeomorphism group of the circle . Comment. Math. Helv. 78, 787–804 (2003). https://doi.org/10.1007/s00014-003-0785-6
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DOI: https://doi.org/10.1007/s00014-003-0785-6