×

zbMATH — the first resource for mathematics

The metric geometry of the manifold of Riemannian metrics over a closed manifold. (English) Zbl 1213.58007
Let \(\mathcal{M}\) be the infinite dimensional Fréchet manifold of all \(C^\infty\) Riemannian metrics on a fixed closed, finite-dimensional, orientable manifold \(M\). This manifold has a natural weak Riemannian metric called the \(L^2\) metric because it induces an \(L^2\)-type scalar product on the tangent spaces of \(\mathcal{M}\). The main result of the paper is that this weak metric induces a metric space structure on \(\mathcal{M}\).
This result is non-trivial because, a priori, the distance function induced by a weak metric on a Fréchet manifold is only a pseudometric, meaning that the distance between some points can be zero. Thus, in their work motivated by research in computer vision, P. W. Michor and D. Mumford found examples of weak Riemannian metrics on Fréchet manifolds for which the distance between any two points is zero [Doc. Math., J. DMV 10, 217–245 (2005; Zbl 1083.58010)].
A motivation for studying the metrics on the manifold \(\mathcal{M}\) comes from Teichmüller theory. Indeed, if \(M\) is a Riemann surface of genus greater than one, then the Techmüller space \(\mathcal{T}\) of \(M\) is a quotient of the submanifold of \(\mathcal{M}\) which consists of hyperbolic metrics. It can be shown that the \(L^2\) metric on \(\mathcal{M}\) descends to \(\mathcal{T}\) and is isometric (up to a constant scalar factor) to the Weil-Petersson metric on \(\mathcal{T}\).
An important ingredient of the proof of the main result is an explicit relation between the induced distance between two metrics in \(\mathcal{M}\) and the corresponding volume forms on the initial manifold \(M\).

MSC:
58D17 Manifolds of metrics (especially Riemannian)
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Clarke, B.: The completion of the manifold of Riemannian metrics. preprint. arXiv:0904.0177v1 · Zbl 1284.58005
[2] Clarke, B.: The completion of the manifold of Riemannian metrics with respect to its L 2 metric. Ph.D. thesis, University of Leipzig. arXiv:0904.0159v1 (2009) · Zbl 1183.53003
[3] Constantin A., Kolev B.: Geodesic flow on the diffeomorphism group of the circle. Comment. Math. Helv. 78, 787–804 (2003) · Zbl 1037.37032 · doi:10.1007/s00014-003-0785-6
[4] DeWitt B.S.: Quantum theory of gravity. I. The canonical theory. Phys. Rev. 160(5), 1113–1148 (1967) · Zbl 0158.46504 · doi:10.1103/PhysRev.160.1113
[5] Ebin, D.G.: The manifold of Riemannian metrics, Global analysis. In: Chern, S.-S., Smale, S. (eds.) Proceedings of Symposia in Pure Mathematics, vol. 15, pp. 11–40. American Mathematical Society, Providence (1970) · Zbl 0205.53702
[6] 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
[7] Gil-Medrano, O., Michor, P.W.: The Riemannian manifold of all Riemannian metrics. Q. J. Math. Oxf. Ser. (2) 42(166), 183–202. arXiv:math/9201259 (1991) · Zbl 0739.58010
[8] Hamilton R.S.: The inverse function theorem of Nash and Moser. Bull. Am. Math. Soc. 7(1), 65–222 (1982) · Zbl 0499.58003 · doi:10.1090/S0273-0979-1982-15004-2
[9] Klingenberg, W.P.A.: Riemannian geometry. De Gruyter Studies in Mathematics, 2nd edn, vol. 1. Walter de Gruyter and Co., Berlin (1995) · Zbl 0911.53022
[10] Lang S.: Differential and Riemannian Manifolds. Graduate Texts in Mathematics, 3rd edn., vol. 160. Springer-Verlag, New York (1995) · Zbl 0824.58003
[11] Michor P.W., Mumford D.: Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Doc. Math. 10, 217–245 (2005) arXiv:math/0409303 · Zbl 1083.58010
[12] Michor P.W., Mumford D.: Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. (JEMS) 8(1), 1–48 (2006) arXiv:math.DG/0312384 · Zbl 1101.58005 · doi:10.4171/JEMS/37
[13] Schaefer H.H.: Topological Vector Spaces. Graduate Texts in Mathematics, 2nd ed., vol. 3. Springer, New York (1999) · Zbl 0983.46002
[14] 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.