## 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)

### Citations:

Zbl 0509.53034; Zbl 0194.529; Zbl 0524.53025
Full Text:

### References:

 [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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.