On completeness of groups of diffeomorphisms

Abstract

We study completeness properties of the Sobolev diffeomorphism groups ${\mathcal{D}}^s(M)$ endowed with strong right-invariant Riemannian metrics when $M$ is ${\mathbb{R}}^d$ or a compact manifold without boundary. We prove that for $s>$ dim $(M)/2+1$, the group ${\mathcal{D}}^s(M)$ is geodesically and metrically complete and any two diffeomorphisms in the same component can be joined by a minimal geodesic. We then present the connection between the Sobolev diffeomorphism group and the large deformation matching framework in order to apply our results to diffeomorphic image matching.