The Completion of the Manifold of Riemannian Metrics with Respect to its $L^2$ Metric
| dc.creator | Clarke, Brian | |
| dc.date | 2009-04-01 | |
| dc.date.accessioned | 2026-07-07T12:59:04Z | |
| dc.date.available | 2026-07-07T12:59:04Z | |
| dc.description | This is the author's Ph.D. thesis, submitted to the University of Leipzig. It deals with the $L^2$ 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 manifold of metrics with respect to the $L^2$ metric. The primary motivation for studying this problem comes from Teichmueller 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 Teichmueller space with respect to a class of metrics that generalize the Weil-Petersson metric. We also prove that the $L^2$ metric induces a metric space structure on the manifold of metrics. 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. | |
| dc.description | Ph.D. Thesis, University of Leipzig 133 pages, 6 figures | |
| dc.identifier | https://arxiv.org/abs/0904.0159 | |
| dc.identifier | http://arxiv.org/abs/0904.0159 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/225459 | |
| dc.subject | Differential Geometry | |
| dc.subject | 58D17 (Primary) 58B20 (Secondary) | |
| dc.title | The Completion of the Manifold of Riemannian Metrics with Respect to its $L^2$ Metric | |
| dc.type | text |