Gromov's compactness theorem (geometry)

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This article is about Gromov's compactness theorem in Riemannian geometry. For Gromov's compactness theorem in symplectic topology, see Gromov's compactness theorem (topology).

In Riemannian geometry, Gromov's (pre)compactness theorem states that the set of Riemannian manifolds of a given dimension, with Ricci curvaturec and diameterD is relatively compact in the Gromov–Hausdorff metric.[1][2] It was proved by Mikhail Gromov.[2][3]

This theorem is a generalization of the Myers's theorem.[4]


  1. ^ Chow, Bennett (2010), The Ricci Flow: Techniques and Applications. Geometric-analytic aspects, Part 3, Mathematical surveys and monographs, American Mathematical Society, p. 396, ISBN 9780821875445 .
  2. ^ a b Bär, Christian; Lohkamp, Joachim; Schwarz, Matthias (2011), Global Differential Geometry, Springer Proceedings in Mathematics 17, Springer, p. 94, ISBN 9783642228421 .
  3. ^ Gromov, Mikhael (1981), Structures métriques pour les variétés riemanniennes, Textes Mathématiques [Mathematical Texts] 1, Paris: CEDIC, ISBN 2-7124-0714-8, MR 682063 . As cited by Bär, Lohkamp & Schwarz (2011).
  4. ^ Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques (2004), Riemannian Geometry, Universitext, Springer, p. 179, ISBN 9783540204930 .