The Completion of the Manifold of Riemannian Metrics with Respect to its $L^2$ Metric

Abstract

This is the author's Ph.D. thesis, submitted to the University of Leipzig. It deals with the $L^2$ Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold. The main body of the thesis is a description of the completion manifold of metrics with respect to the $L^2$ metric. The primary motivation for studying this problem comes from Teichmueller theory, where similar considerations lead to a completion of the well-known Weil-Petersson metric. We give an application of the main theorem to the completions of Teichmueller space with respect to a class of metrics that generalize the Weil-Petersson metric. We also prove that the $L^2$ metric induces a metric space structure on the manifold of metrics. As the $L^2$ metric is a weak Riemannian metric, this fact does not follow from general results. In addition, we prove several results on the exponential mapping and distance function of a weak Riemannian metric on a Hilbert/Frechet manifold. The statements are analogous to, but weaker than, what is known in the case of a Riemannian metric on a finite-dimensional manifold or a strong Riemannian metric on a Hilbert manifold.

arXiv commentPh.D. Thesis, University of Leipzig 133 pages, 6 figures

References in corpus (2)

• An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach
• Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms

Cited by in corpus (7)

• The Riemannian L^2 topology on the manifold of Riemannian metrics
• The Metric Geometry of the Manifold of Riemannian Metrics over a Closed Manifold
• Short survey on the existence of slices for the space of Riemannian metrics
• A metric on the space of neutron star models in general relativity and modified gravity
• Ricci Curvature, Reeb Flows and Contact 3-Manifolds
• The Completion of the Manifold of Riemannian Metrics
• Geodesics, distance, and the CAT(0) property for the manifold of Riemannian metrics