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

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

2012-02-16

arxiv.org/abs/1202.3677v1

Mathematics

Differential Geometry

28 pages

Scientific paper

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 $B(M,N)$.

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