Geodesic flow on the diffeomorphism group of the circle

We show that certain right-invariant metrics endow theinfinite-dimensional Lie group of all smoothorientation-preserving diffeomorphisms of the circle with aRiemannian structure. The study of the Riemannian exponential mapallows us to prove infinite-dimensional counterparts of resultsfrom classical Riemannian geometry: the Riemannian exponential map isa smooth local diffeomorphism and the length-minimizing property of the geodesics holds.