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.
Cite this article
Adrian Constantin, Boris Kolev, Geodesic flow on the diffeomorphism group of the circle. Comment. Math. Helv. 78 (2003), no. 4, pp. 787–804