The Riemannian \(L^{2}\) topology on the manifold of Riemannian metrics. (English) Zbl 1213.58008
This paper builds on and develops the results of the previous work by the author [Calc. Var. Partial Differ. Equ. 39, No. 3–4, 533–545 (2010; Zbl 1213.58007)]. The object of study is the manifold \(\mathcal{M}\) of all Riemannian metrics on a closed, finite-dimensional manifold \(M\). In this article, the author gives a simplified description of the topology induced by the distance function of the \(L^2\) metric on \(\mathcal{M}\), as well as on its completion \(\overline{\mathcal{M}}\). It turns out that this \(L^2\) topology on the tangent spaces gives rise to an \(L^1\)-type topology on the manifold of metrics itself. The author studies this new topology and its completion, which agrees homeomorphically with the completion of the \(L^{2}\) metric. He also gives a new practical criterion for convergence (with respect to the \(L^{2}\) metric) in the manifold of metrics.
Reviewer: Mikhail Belolipetsky (Durham)
MSC:
| 58D17 | Manifolds of metrics (especially Riemannian) |
| 58B20 | Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds |
Keywords:
manifold of Riemannian metrics; superspace; manifold of Riemannian structures; \(L^{2}\) metricCitations:
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