Almost local metrics on shape space of hypersurfaces in n-space

Abstract. This paper extends parts of the results from [17] for plane curves to the case of hypersurfaces in Rn. Let M be a compact connected oriented n − 1 dimensional manifold without boundary like S2 or the torus S1 × S1. Then shape space is either the manifold of submanifolds of Rn of type M, or the orbifold of immersions from M to Rn modulo the group of diffeomorphisms of M. We investigate almost local Riemannian metrics on shape space: These are induced by metrics of the following form on the space of immersions: Gf (h, k) = M Φ(Vol(M),Tr(L))ḡ(h, k) vol(f∗ḡ) where g ̄ is the standard metric on Rn, f∗g ̄ is the induced metric on M, h, k ∈ C∞(M,Rn) are tangent vectors at f to the space of embeddings or immersions, where Φ: R2 → R>0 is a suitable smooth function, Vol(M) = M vol(f ∗ḡ