| 词条 | Gromov's compactness theorem (geometry) |
| 释义 |
In Riemannian geometry, Gromov's (pre)compactness theorem states that the set of compact Riemannian manifolds of a given dimension, with Ricci curvature ≥ c and diameter ≤ D 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] References1. ^{{citation|title=The Ricci Flow: Techniques and Applications. Geometric-analytic aspects, Part 3|series=Mathematical surveys and monographs|first=Bennett|last=Chow|publisher=American Mathematical Society|year=2010|isbn=9780821875445|page=396|url=https://books.google.com/books?id=RsQuPui9i3EC&pg=PA396}}. {{differential-geometry-stub}}2. ^1 {{citation|title=Global Differential Geometry|volume=17|series=Springer Proceedings in Mathematics|first1=Christian|last1=Bär|first2=Joachim|last2=Lohkamp|first3=Matthias|last3=Schwarz|publisher=Springer|year=2011|isbn=9783642228421|page=94|url=https://books.google.com/books?id=SipB51TUH8EC&pg=PA94}}. 3. ^{{citation | last = Gromov | first = Mikhael | isbn = 2-7124-0714-8 | location = Paris | mr = 682063 | publisher = CEDIC | series = Textes Mathématiques [Mathematical Texts] | title = Structures métriques pour les variétés riemanniennes | volume = 1 | year = 1981}}. As cited by {{harvtxt|Bär|Lohkamp|Schwarz|2011}}. 4. ^{{citation|title=Riemannian Geometry|series=Universitext|first1=Sylvestre|last1=Gallot|first2=Dominique|last2=Hulin|first3=Jacques|last3=Lafontaine|publisher=Springer|year=2004|isbn=9783540204930|page=179|url=https://books.google.com/books?id=6F4Umpws_gUC&pg=PA179}}. 1 : Theorems in Riemannian geometry |
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