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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
21/2009

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

Brian Clarke

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.

Received:
07.04.09
Published:
09.04.09
MSC Codes:
58D17, 58B20

Related publications

inJournal
2010 Repository Open Access
Brian Clarke

The metric geometry of the manifold of Riemannian metrics over a closed manifold

In: Calculus of variations and partial differential equations, 39 (2010) 3/4, pp. 533-545