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 ${\mbox{diam}}(M,g_{\varepsilon })\leq 1$ 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 $\leq 1$ 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.

References

• Gromov, M. (1978), "Almost flat manifolds", Journal of Differential Geometry, 13 (2): 231–241, MR 0540942.
• Ruh, Ernst A. (1982), "Almost flat manifolds", Journal of Differential Geometry, 17 (1): 1–14, MR 0658470.