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.


37K65 Hamiltonian systems on groups of diffeomorphisms and on manifolds of mappings and metrics
35Q35 PDEs in connection with fluid mechanics
58B25 Group structures and generalizations on infinite-dimensional manifolds
53D25 Geodesic flows in symplectic geometry and contact geometry
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