Gromov's compactness theorem (geometry)

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In Riemannian geometry, Gromov's (pre)compactness theorem states that the set of compact 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 in 1981.[2][3]

This theorem is a generalization of 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 .