Abstract
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.
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References
Clarke, B.: The completion of the manifold of Riemannian metrics. preprint. arXiv:0904.0177v1
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)
Constantin A., Kolev B.: Geodesic flow on the diffeomorphism group of the circle. Comment. Math. Helv. 78, 787–804 (2003)
DeWitt B.S.: Quantum theory of gravity. I. The canonical theory. Phys. Rev. 160(5), 1113–1148 (1967)
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)
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)
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)
Hamilton R.S.: The inverse function theorem of Nash and Moser. Bull. Am. Math. Soc. 7(1), 65–222 (1982)
Klingenberg, W.P.A.: Riemannian geometry. De Gruyter Studies in Mathematics, 2nd edn, vol. 1. Walter de Gruyter and Co., Berlin (1995)
Lang S.: Differential and Riemannian Manifolds. Graduate Texts in Mathematics, 3rd edn., vol. 160. Springer-Verlag, New York (1995)
Michor P.W., Mumford D.: Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Doc. Math. 10, 217–245 (2005) arXiv:math/0409303
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
Schaefer H.H.: Topological Vector Spaces. Graduate Texts in Mathematics, 2nd ed., vol. 3. Springer, New York (1999)
Tromba A.J.: Teichmüller Theory in Riemannian Geometry. Birkhäuser, Basel (1992)
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Communicated by J. Jost.
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Clarke, B. The metric geometry of the manifold of Riemannian metrics over a closed manifold. Calc. Var. 39, 533–545 (2010). https://doi.org/10.1007/s00526-010-0323-5
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DOI: https://doi.org/10.1007/s00526-010-0323-5