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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1$\begingroup$ I would like to add Ruh's extension to Gromov's paper "Almost flat manifolds" J. Differential Geom. 17 (1982), no. 1, 1–14. $\endgroup$– J. GECommented Feb 5, 2014 at 20:30
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2 Answers
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M. Gromov, Almost flat manifolds, J. Differential Geometry 13 (1978) 231-241
online at http://seven.ihes.fr/~gromov/PDF/3[20].pdf
answered Jan 27, 2014 at 8:55
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The book "Gromov's almost flat manifolds" by Buser and Karcher is a nice exposition of Gromov's result.
