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On the number of diffeomorphism classes in a certain class of Riemannian manifolds. (English) Zbl 0566.53038

Cheeger’s finiteness theorem asserts that there are only finitely many diffeomorphism classes in the class of compact Riemannian manifolds M of dimension n, such that the curvature of M is between two given bounds - \(\Lambda\) \({}^ 2_ 1\) and \(+ \Lambda^ 2_ 2\) and diameter(M)\(\leq D\), volume(M)\(\geq V\), where D and V are given. Extensions of this result to complete manifolds are discussed.
Reviewer: W.Ballmann

MSC:

53C20 Global Riemannian geometry, including pinching
57R55 Differentiable structures in differential topology
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References:

[1] (1981)
[2] J. Differential Geom 13 pp 231– (1978) · Zbl 0432.53020
[3] DOI: 10.1007/BF01226419 · Zbl 0373.53018
[4] Comparison theorems in Riemannian geometry (1975) · Zbl 0309.53035
[5] DOI: 10.2307/2373498 · Zbl 0194.52902
[6] DOI: 10.2307/2373353 · Zbl 0183.50301
[7] Ann. Sci. Ecole Norm. Sup 11 pp 451– (1978) · Zbl 0416.53027
[8] Gromov’s almost flat manifolds (1981) · Zbl 0459.53031
[9] Geometry of manifolds (1964)
[10] DOI: 10.1007/BF01899493 · Zbl 0166.17601
[11] Osaka J. Math 3 pp 65– (1966)
[12] DOI: 10.2969/jmsj/03030533 · Zbl 0397.53039
[13] Comparison and finiteness theorems for Riemannian manifolds (1967)
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