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

TL;DR

This paper develops a geometric framework using Sobolev metrics on diffeomorphism groups to analyze the space of submanifolds, deriving formulas for geodesics and curvature to understand their geometric properties.

Contribution

It introduces a general class of Sobolev-type metrics on diffeomorphism groups and computes the induced geometry on submanifold spaces, including geodesics and curvature formulas.

Findings

Derived explicit formulas for geodesics on submanifold spaces.

Computed sectional curvature using a covariant curvature formula.

Clarified the geometric structure of submanifold spaces under Sobolev metrics.

Abstract

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 $B(M,N)$.