Almost flat manifold

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.