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


53C20 Global Riemannian geometry, including pinching
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58D17 Manifolds of metrics (especially Riemannian)
Full Text: DOI


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