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:
Zbl 1213.58007References:
[1] | Bogachev V.I.: Measure Theory, vols. I & II. Springer, Berlin (2007) · Zbl 1120.28001 |
[2] | Bourguignon J.-P.: Une stratification de l’espace des structures riemanniennes. Compos. Math. 30(1), 1–41 (1975) · Zbl 0301.58015 |
[3] | Clarke, B.: The completion of the manifold of Riemannian metrics with respect to its L 2 metric. Ph.D. Thesis, University of Leipzig, 2009, http://arxiv.org/abs/0904.0159v1 |
[4] | Clarke, B.: The completion of the manifold of Riemannian metrics, preprint, http://arxiv.org/abs/0904.0177v1 |
[5] | Clarke, B.: The metric geometry of the manifold of Riemannian metrics. Calc. Var. Partial Differential Equations, pp.1–13. Springer, Berlin/Heidelberg (2010) |
[6] | Ebin, D.G.: The manifold of Riemannian metrics. In: Chern, S.-S., Smale, S. (eds.) Global Analysis. Proceedings of Symposia in Pure Mathematics, vol. 15, pp. 11–40. American Mathematical Society, Providence, RI (1970) · Zbl 0205.53702 |
[7] | Freed D.S., Groisser D.: The basic geometry of the manifold of Riemannian metrics and of its quotient by the diffeomorphism group. Michigan Math. J. 36, 323–344 (1989) · Zbl 0694.58008 · doi:10.1307/mmj/1029004004 |
[8] | Gil-Medrano, O., Michor, P.W.: The Riemannian manifold of all Riemannian metrics. Q. J. Math. Oxford Ser. (2) 42(166), 183–202 (1991), http://arxiv.org/abs/arXiv:math/9201259 · Zbl 0739.58010 |
[9] | Gromov M.: Metric Structures for Riemannian and Non-Riemannian Spaces. Birkhäuser, Boston (2007) · Zbl 1113.53001 |
[10] | Michor, P.W., Mumford, D.: Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Doc. Math. 10, 217–245 (2005), http://arxiv.org/abs/arXiv:math/0409303 · Zbl 1083.58010 |
[11] | Michor, P.W., Mumford, D.: Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. (JEMS) 8(1), 1–48 (2006), http://arxiv.org/abs/math.DG/0312384 · Zbl 1101.58005 |
[12] | Rana I.K.: An Introduction to Measure and Integration, 2nd ed. Graduate Studies in Mathematics, vol. 45. American Mathematical Society, Providence, RI (2002) · Zbl 1003.28001 |
[13] | Tromba A.J.: Teichmüller Theory in Riemannian Geometry. Birkhäuser, Basel (1992) · Zbl 0785.53001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.