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

  • Brian ClarkeEmail author
Article

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/Fréchet 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.

Mathematics Subject Classification (2000)

58D17 (Primary) 58B20 (Secondary) 

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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