The Completion of the Manifold of Riemannian Metrics with Respect to its L² Metric

Brian Clarke

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Submission date: 07. Apr. 2009
Pages: 135
MSC-Numbers: 58D17, 58B20
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Abstract:
This is the author's Ph.D. thesis, submitted to the University of Leipzig. It deals with the 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 of the manifold of metrics with respect to the metric. The primary motivation for studying this problem comes from Teichmüller 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 Teichmüller space with respect to a class of metrics that generalize the Weil-Petersson metric.

We also prove that the metric induces a metric space structure on the manifold of metrics. As the 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.