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

Given a finite-dimensional manifold $N$, the group $\operatorname{Diff}_{\mathcal S}(N)$ of diffeomorphisms diffeomorphism 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 $\operatorname{dim} M<\operatorname{dim} N$. Given the right-invariant weak Riemannian metric on $\operatorname{Diff}_{\mathcal S}(N)$ induced by a quite general operator $L\colon \mathfrak X_{\mathcal S}(N)\to \Gamma(T^*N\otimes\operatorname{vol}(N))$, 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 $B(M,N)$.
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