Sobolev Metrics on Diffeomorphism Groups and the Derived Geometry of Spaces of Submanifolds

Given a finite dimensional manifold N , the group DiffS(N) of diffeomorphism of N which fall suitably rapidly to the identity, acts on the manifold B(M,N) of submanifolds on N of diffeomorphism type M where M is a compact manifold with dimM < dimN . For a right invariant weak Riemannian metric on DiffS(N) induced by a quite general operator L : XS(N)→ Γ(T ∗N⊗vol(N)), we consider the induced weak Riemannian metric on B(M,N) and we compute its geodesics and sectional curvature. For that we derive a covariant formula for curvature in finite and infinite dimensions, we show how it makes O’Neill’s formula very transparent, and we use it finally to compute sectional curvature on B(M,N).