Almost flat manifold

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

Given n, there is a positive number \varepsilon_n>0 such that if an n-dimensional manifold admits an \varepsilon_n-flat metric with diameter \le 1 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, M is almost flat if and only if it is 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.