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In his talk Misha Gromov- How does he do it, Jeff Cheeger mentions a theorem of Gromov proved sometime in the early 70's. Theorem: Every manifold admitting a sequence of metrics such that the diameter and curvature go to zero is finitely covered by a nilmanifold. Further, he says that the proof is not easy even today; something usually associated with Gromov! It would be great to read the original paper, and even better if some book contains it. Please help me find them. Thanks.

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I would like to add Ruh's extension to Gromov's paper "Almost flat manifolds" J. Differential Geom. 17 (1982), no. 1, 1–14. –  J. GE Feb 5 at 20:30

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up vote 6 down vote accepted

M. Gromov, Almost flat manifolds, J. Differential Geometry 13 (1978) 231-241

online at http://seven.ihes.fr/~gromov/PDF/3[20].pdf

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The book "Gromov's almost flat manifolds" by Buser and Karcher is a nice exposition of Gromov's result.

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