# Gromov's compactness theorem (Geometrie)

Gromov's compactness theorem (Geometrie) Not to be confused with Gromov's compactness theorem in symplectic geometry.

In the mathematical field of metric geometry, Mikhael Gromov proved a fundamental compactness theorem for sequences of metric spaces. In the special case of Riemannian manifolds, the key assumption of his compactness theorem is automatically satisfied under an assumption on Ricci curvature. These theorems have been widely used in the fields of geometric group theory and Riemannian geometry.

Metric compactness theorem The Gromov–Hausdorff distance defines a notion of distance between any two metric spaces, thereby setting up the concept of a sequence of metric spaces which converges to another metric space. This is known as Gromov–Hausdorff convergence. Gromov found a condition on a sequence of compact metric spaces which ensures that a subsequence converges to some metric space relative to the Gromov–Hausdorff distance:[1] Lassen (Xi, di) be a sequence of compact metric spaces with uniformly bounded diameter. Suppose that for every positive number ε there is a natural number N and, for every i, the set Xi can be covered by N metric balls of radius ε. Then the sequence (Xi, di) has a subsequence which converges relative to the Gromov–Hausdorff distance.

The role of this theorem in the theory of Gromov–Hausdorff convergence may be considered as analogous to the role of the Arzelà–Ascoli theorem in the theory of uniform convergence.[2] Gromov first formally introduced it in his 1981 resolution of the Milnor–Wolf conjecture in the field of geometric group theory, where he applied it to define the asymptotic cone of certain metric spaces.[3] These techniques were later extended by Gromov and others, using the theory of ultrafilters.[4] Riemannian compactness theorem Specializing to the setting of geodesically complete Riemannian manifolds with a fixed lower bound on the Ricci curvature, the crucial covering condition in Gromov's metric compactness theorem is automatically satisfied as a corollary of the Bishop–Gromov volume comparison theorem. As such, es folgt dem:[5] Consider a sequence of closed Riemannian manifolds with a uniform lower bound on the Ricci curvature and a uniform upper bound on the diameter. Then there is a subsequence which converges relative to the Gromov–Hausdorff distance.

The limit of a convergent subsequence may be a metric space without any smooth or Riemannian structure.[6] This special case of the metric compactness theorem is significant in the field of Riemannian geometry, as it isolates the purely metric consequences of lower Ricci curvature bounds.

References ^ Bridson & Haefliger 1999, Satz 5.41; Burago, Burago & Ivanov 2001, Satz 7.4.15; Gromov 1981, Abschnitt 6; Gromov 1999, Vorschlag 5.2; Petersen 2016, Vorschlag 11.1.10. ^ Villani 2009, p. 754. ^ Gromov 1981, Abschnitt 7; Gromov 1999, Paragraph 5.7. ^ Bridson & Haefliger 1999, Definition 5.50; Gromov 1993, Abschnitt 2. ^ Gromov 1999, Satz 5.3; Petersen 2016, Logische Folge 11.1.13. ^ Gromov 1999, Paragraph 5.5.

Quellen.

Bridson, Martin R.; Haefliger, André (1999). Metric spaces of non-positive curvature. Grundlehren der mathematischen Wissenschaften. Vol. 319. Berlin: Springer-Verlag. doi:10.1007/978-3-662-12494-9. ISBN 3-540-64324-9. HERR 1744486. Zbl 0988.53001. Burago, Dmitri; Burago, Juri; Iwanow, Sergei (2001). A course in metric geometry. Studium der Mathematik. Vol. 33. Vorsehung, RI: Amerikanische Mathematische Gesellschaft. doi:10.1090/gsm/033. ISBN 0-8218-2129-6. HERR 1835418. Zbl 0981.51016. (Erratum: [1]) Gromov, Mikhael (1981). "Groups of polynomial growth and expanding maps". Publications Mathématiques de l'Institut des Hautes Études Scientifiques. 53: 53–73. doi:10.1007/BF02698687. HERR 0623534. Zbl 0474.20018. Gromov, M. (1993). "Asymptotic invariants of infinite groups". In Niblo, Graham A.; Rolle, Martin A. (Hrsg.). Geometrische Gruppentheorie. Vol. 2. Symposium held at Sussex University (Sussex, Juli 1991). Vorlesungsreihe der London Mathematical Society. Cambridge: Cambridge University Press. pp. 1–295. ISBN 0-521-44680-5. HERR 1253544. Zbl 0841.20039. Gromov, Mischa (1999). Metric structures for Riemannian and non-Riemannian spaces. Progress in Mathematics. Vol. 152. Translated by Bates, Sean Michael. With appendices by M. Katz, P. Pansu, and S. Semmes. (Based on the 1981 French original ed.). Boston, MA: Birkhäuser Boston, Inc. doi:10.1007/978-0-8176-4583-0. ISBN 0-8176-3898-9. HERR 1699320. Zbl 0953.53002. Petersen, Peter (2016). Riemannsche Geometrie. Abschlusstexte in Mathematik. Vol. 171 (Dritte Auflage von 1998 original ed.). Springer, Cham. doi:10.1007/978-3-319-26654-1. ISBN 978-3-319-26652-7. HERR 3469435. Zbl 1417.53001. Villans, Cédric (2009). Optimal transport. Old and new. Grundlehren der mathematischen Wissenschaften. Vol. 338. Berlin: Springer-Verlag. doi:10.1007/978-3-540-71050-9. ISBN 978-3-540-71049-3. HERR 2459454. Zbl 1156.53003. vte Verteiler ausblenden (Glossar) Grundkonzepte Topologische Mannigfaltigkeit AtlasDifferentiable/Glatte Mannigfaltigkeit DifferentialstrukturGlatter AtlasUntermannigfaltigkeitRiemannsche MannigfaltigkeitGlatte KarteSubmersionPushforwardTangentenraumDifferentialformVektorfeld Hauptergebnisse (aufführen) Atiyah–Singer indexDarboux’sDe Rham’sFrobeniusGeneralized StokesHopf–RinowNoether’sSard’sWhitney embedding Maps CurveDiffeomorphism LocalGeodesicExponential map in Lie theoryFoliationImmersionIntegralkurveLie-AbleitungSectionSubmersion Types of manifolds Closed(Fast) Komplex(Fast) KontaktFiberedFinslerFlatG-StrukturHadamardHermitianHyperbolicKählerKenmotsuLie-Gruppe Lie-AlgebraManifold with borderOrientedParallelizablePoissonPrimeQuaternionicHypercomplex(Pseudo−, Unter−) RiemannianRizza(Fast) SymplecticTame Tensors Vectors DistributionLie BracketPushforwardTangential space bundleTorsionVector fieldVector flow Covectors Closed/ExactCovariant ableiteCotangent space bundleDe Rham cohomologyDifferential form Vector-valuedExterior derivativeInterior productPullbackRicci Curvature flowRiemann curvature tensorTensor field densityVolume formWedge product Bundles AdjointAffineAssociatedCotangentDualFiber(Co) FibrationJetLie-Algebra(Stabil) NormalPrincipalSpinorSubbundleTangensTensorVector Connections AffineCartanEhresmannFormGeneralizedKoszulLevi-CivitaPrincipalVectorParallel transport Related Klassifikation von MannigfaltigkeitenEichtheorieGeschichteMorsetheorieBewegender RahmenSingularitätstheorie Verallgemeinerungen Banach-MannigfaltigkeitDiffeologieDiffietyFréchet-MannigfaltigkeitK-TheoryOrbifoldSekundärrechnung über kommutative AlgebrenGarbeStratifoldSupermanifoldSupermanifoldSupermanifold: Differential geometryTheorems in Riemannian geometry

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