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Sobolev Metrics on Diffeomorphism Groups and the Derived Geometry of Spaces of Submanifolds by Micheli, Mario; Michor, Peter W.; Mumford, David is a scholarly article available to read on EtoBox.

What is Sobolev Metrics on Diffeomorphism Groups and the Derived Geometry of Spaces of Submanifolds about?

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

Author

Micheli, Mario; Michor, Peter W.; Mumford, David

Published

2012

Language

EN