The metric geometry of the manifold of Riemannian metrics over a closed manifold

  • Brian ClarkeEmail author


We prove that the L 2 Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold induces a metric space structure. 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/Fréchet 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.

Mathematics Subject Classification (2000)

58D17 (Primary) 58B20 (Secondary) 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Clarke, B.: The completion of the manifold of Riemannian metrics. preprint. arXiv:0904.0177v1Google Scholar
  2. 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)Google Scholar
  3. 3.
    Constantin A., Kolev B.: Geodesic flow on the diffeomorphism group of the circle. Comment. Math. Helv. 78, 787–804 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    DeWitt B.S.: Quantum theory of gravity. I. The canonical theory. Phys. Rev. 160(5), 1113–1148 (1967)zbMATHCrossRefGoogle Scholar
  5. 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)Google Scholar
  6. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 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)Google Scholar
  8. 8.
    Hamilton R.S.: The inverse function theorem of Nash and Moser. Bull. Am. Math. Soc. 7(1), 65–222 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Klingenberg, W.P.A.: Riemannian geometry. De Gruyter Studies in Mathematics, 2nd edn, vol. 1. Walter de Gruyter and Co., Berlin (1995)Google Scholar
  10. 10.
    Lang S.: Differential and Riemannian Manifolds. Graduate Texts in Mathematics, 3rd edn., vol. 160. Springer-Verlag, New York (1995)Google Scholar
  11. 11.
    Michor P.W., Mumford D.: Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Doc. Math. 10, 217–245 (2005) arXiv:math/0409303zbMATHMathSciNetGoogle Scholar
  12. 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/0312384zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Schaefer H.H.: Topological Vector Spaces. Graduate Texts in Mathematics, 2nd ed., vol. 3. Springer, New York (1999)Google Scholar
  14. 14.
    Tromba A.J.: Teichmüller Theory in Riemannian Geometry. Birkhäuser, Basel (1992)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityStanfordUSA

Personalised recommendations