The Metric Geometry of the Manifold of Riemannian Metrics over a Closed Manifold

Abstract

We prove that the L^2 Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold induces a metric space structure. 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 comment14 pages, no figures; Corrected typo in Theorem 10

References in corpus (2)

• An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach
• The Completion of the Manifold of Riemannian Metrics with Respect to its $L^2$ Metric

Cited by in corpus (4)

• The Riemannian L^2 topology on the manifold of Riemannian metrics
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• The Completion of the Manifold of Riemannian Metrics
• Geodesics, distance, and the CAT(0) property for the manifold of Riemannian metrics