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In mathematics, a smooth compact manifold M is called almost flat if for any <math>\epsilon>0 <math> there is a Riemannian metric <math>g_\epsilon <math> on M such that <math>\mbox{diam}(M,g_\epsilon)\le 1 <math> and
<math>g_\epsilon <math> is <math>\epsilon<math>-flat, i.e. for sectional curvature of <math>K_{g_\epsilon}<math> we have <math>|K_{g_\epsilon}|<\epsilon<math>.
In fact, given n, there is a positive number <math>\epsilon_n>0 <math> such that if a n-dimensional manifold admits an <math>\epsilon_n<math>-flat metric with diameter <math>\le 1 <math> then it is almost flat.
According to the Gromov-Ruh theorem, M is almost flat if and only if it is infranil. In particular, it is a finite factor of a nill manifold, i.e. a total space of a oriented circle bundle over a oriented circle bundle over ... over a circle.
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