Fractional Sobolev Metrics on Spaces of Immersed Curves

Abstract

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(S1 , R 𝑑 ) and on its Sobolev completions ℐ 𝑞 (S1 , R 𝑑 ). We prove local well-posedness of the geodesic equations both on the Banach manifold ℐ 𝑞 (S1 , R 𝑑 ) and on the Fr´echetmanifold Imm(S1 , R 𝑑 ) provided the order of the metric is greater or equal to one. In addition we show that the 𝐻𝑠 -metric induces a strong Riemannian metric on the Banach manifold ℐ 𝑠 (S1 , 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.