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Geodesic flow on the diffeomorphism group of the circle. (English) Zbl 1037.37032
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:
37K65Hamiltonian systems on groups of diffeomorphisms and on manifolds of mappings and metrics
35Q35PDEs in connection with fluid mechanics
58B25Group structures and generalizations on infinite-dimensional manifolds
53D25Geodesic flows (symplectic geometry)