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.

Notes

References

• Gromov, M. (1978), "Almost flat manifolds", Journal of Differential Geometry 13 (2): 231–241, MR 540942.
• Ruh, Ernst A. (1982), "Almost flat manifolds", Journal of Differential Geometry 17 (1): 1–14, MR 658470.