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

Given a finite-dimensional manifold N, the group Diff(S)(N) of diffeomorphisms of N which decrease suitably rapidly to the identity, acts on the manifold B(M, N) of submanifolds of N of diffeomorphism-type M, where M is a compact manifold with dim M < dim N. Given the right-invariant weak Riemannian metric on DiffS(N) induced by a quite general operator L: (sic)(S)(N) -> (T* N circle times vol(N)), we consider the induced weak Riemannian metric on B(M, N) 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 B(M, N).