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

Abstract

Given a finite dimensional manifold $N$, the group ${\mathrm{Diff}}_{\mathcal{S}}(N)$ 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 ${\mathrm{Diff}}_{\mathcal{S}}(N)$ induced by a quite general operator $L:{\mathfrak{X}}_{\mathcal{S}}(N)\to \Gamma (T^{\ast }N\otimes \mathrm{vol}(N))$, 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 $B(M,N)$.

28 pages. In this version some misprints corrected

References in corpus (3)

• An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach
• Sectional Curvature in terms of the Cometric, with Applications to the Riemannian Manifolds of Landmarks
• The homogeneous Sobolev metric of order one on diffeomorphism groups on the real line

Cited by in corpus (11)

• Geodesic Completeness for Sobolev Metrics on the Space of Immersed Plane Curves
• Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group. II
• The Lagrangian Radon Transform and the Weil representation
• Manifolds of mappings for continuum mechanics
• The homogeneous Sobolev metric of order one on diffeomorphism groups on the real line
• An introduction to infinite-dimensional differential geometry
• Nonlinear flag manifolds as coadjoint orbits
• Sobolev metrics on spaces of manifold valued curves
• Regularity and completeness of half-Lie groups
• A review of some recent work on hypercyclicity
• The Michor-Mumford conjecture in Hilbertian H-type groups