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Gromov’s convergence theorem and its application. (English) Zbl 0587.53043

The author gives a detailed proof of Gromov’s convergence theorem [Théorème 8.25 in M. Gromov, Structures métriques pour les variétés riemanniennes (1981; Zbl 0509.53034)], which has arisen much interest recently. It says in Gromov’s formulation that in the set of Riemannian manifolds of dimension n, with sectional curvature \(| K| \leq \Lambda^ 2\), diameter bounded from above by D, injectivity radius bounded from below by \(\epsilon\), the Hausdorff and Lipschitz topologies coincide. In particular, if a sequence \(V_ i\) of such manifolds converges to V, then \(V_ i\) is homeomorphic to V for i large. As an application the author proves Cheeger’s finiteness theorem [see J. Cheeger, Am. J. Math. 92, 61-74 (1970; Zbl 0194.529)] and S. Peters [J. Reine Angew. Math. 349, 77-82 (1984; Zbl 0524.53025)] and a differentiable sphere theorem for manifolds of positive Ricci curvature.
Another proof of Gromov’s theorem has been obtained independently by S. Peters [Doctoral thesis, Univ. Bonn (1985)] and R. E. Greene and H. Wu [Lipschitz convergence of Riemannian manifolds].
Reviewer: G.Thorbergsson

MSC:

53C20 Global Riemannian geometry, including pinching
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58D17 Manifolds of metrics (especially Riemannian)
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[1] J. Differential Geom 18 pp 151– (1983) · Zbl 0512.53043
[2] Advanced Studies in Pure Math 3 pp 333–
[3] redige par J. Lafontaine et P. Pansu (1981)
[4] J. Differential Geom 13 pp 231– (1978) · Zbl 0432.53020
[5] DOI: 10.1007/BF01394058 · Zbl 0505.53018
[6] DOI: 10.1007/BF01214381 · Zbl 0329.53035
[7] Advanced studies in Pure Math 3 pp 183–
[8] Comparison Theorems in Riemannian Geometry (1975) · Zbl 0309.53035
[9] DOI: 10.1090/S0002-9947-1983-0682734-1
[10] DOI: 10.2307/2373498 · Zbl 0194.52902
[11] Gorden and Breach science Pub
[12] Soc. Math. France 81 (1983)
[13] Advanced Studies in Pure Math 3 pp 125–
[14] J. reine angew. Math 349 pp 77– (1984)
[15] DOI: 10.2969/jmsj/03030533 · Zbl 0397.53039
[16] Ann. Sci. Ecole Norm. Sup. 4e serie 11 pp 451– (1978)
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