Almost flat manifold

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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 an oriented circle bundle over an oriented circle bundle over ... over a circle.