Editing Almost flat manifold (section)

In mathematics, a smooth [[compact space|compact]] [[manifold]] ''M'' is called '''almost flat''' if for any <math>\varepsilon>0 </math> there is a [[Riemannian metric]] <math>g_\varepsilon </math> on ''M'' such that <math> \mbox{diam}(M,g_\varepsilon)\le 1 </math> and 
<math> g_\varepsilon </math> is <math>\varepsilon</math>-flat, i.e. for the [[sectional curvature]] of <math> K_{g_\varepsilon} </math> we have <math> |K_{g_\epsilon}| < \varepsilon</math>.

Given <math>n</math>, there is a positive number <math>\varepsilon_n>0 </math> such that if an <math>n</math>-dimensional manifold admits an <math>\varepsilon_n</math>-flat metric with [[Diameter of a set|diameter]] <math>\le 1 </math> then it is almost flat. On the other hand, one can fix the bound of sectional curvature and get the diameter going to zero, so the almost-flat manifold is a special case of a [[collapsing manifold]], which is collapsing along all directions.

According to the '''Gromov–Ruh theorem''', <math>M</math> is almost flat if and only if it is [[Glossary of Riemannian and metric geometry#I|infranil]]. In particular, it is a finite factor of a [[nilmanifold]], which is the total space of a principal torus bundle over a principal torus bundle over a torus.