On completeness of groups of diffeomorphisms

We study completeness properties of the Sobolev diffeomorphism groups D^s(M) endowed with strong right-invariant Riemannian metrics when M is R^d or a compact manifold without boundary. We prove that for s > dim (M)/2+1 , the group 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.