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

M. Micheli ; P. W. Michor ; D. Mumford

Izvestiya. Mathematics , Tome 77 (2013) no. 3, pp. 541-570

Résumé

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 $B(M,N)$. Bibliography: 15 titles.

Détail

Keywords: robust infinite-dimensional weak Riemannian manifolds, curvature in terms of the cometric, right-invariant Sobolev metrics on diffeomorphism groups, manifold of submanifolds.
Mots-clés : O'Neill's formula