Fractional Sobolev metrics on spaces of immersed curves

Motivated by applications in the field of shape analysis, we study reparametrization invariant, fractional order Sobolev-type metrics on the space of smooth regular curves Imm ( S 1 , R d ) and on its Sobolev completions I q ( S 1 , R d ) . We prove local well-posedness of the geodesic equations both on the Banach manifold I q ( S 1 , R d ) and on the Fréchet-manifold Imm ( S 1 , R d ) provided the order of the metric is greater or equal to one. In addition we show that the H s -metric induces a strong Riemannian metric on the Banach manifold I s ( S 1 , R d ) of the same order s , provided s > 3 2 . These investigations can be also interpreted as a generalization of the analysis for right invariant metrics on the diffeomorphism group.