Sobolev metrics on diffeomorphism groups and the derived geometry of spaces of submanifolds

Abstract

Given a finite-dimensional manifold , the group of diffeomorphisms of which decrease suitably rapidly to the identity, acts on the manifold of submanifolds of of diffeomorphism-type , where is a compact manifold with . Given the right-invariant weak Riemannian metric on induced by a quite general operator , we consider the induced weak Riemannian metric on and compute its geodesics and sectional curvature. To do this, we derive a covariant formula for the curvature in finite and infinite dimensions, we show how it makes O'Neill's formula very transparent, and we finally use it to compute the sectional curvature on .