Doctoral thesis 3/2009

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
Download full preprint: PDF (1086 kB)

This is the author's Ph.D. thesis, submitted to the University of Leipzig. It deals with the formula3 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 formula3 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 formula3 metric induces a metric space structure on the manifold of metrics. As the formula3 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.

04.09.2019, 14:40