Almost flat manifold

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

Given n, there is a positive number ${\displaystyle \varepsilon _{n}>0}$ such that if an n-dimensional manifold admits an ${\displaystyle \varepsilon _{n}}$-flat metric with diameter ${\displaystyle \leq 1}$ 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.