Almost flat manifold

From Wikipedia, the free encyclopedia

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

In fact, given n, there is a positive number $ε n > 0$ such that if a n-dimensional manifold admits an $ε n$ -flat metric with diameter $\le 1$ then it is almost flat.

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, i.e. a total space of an oriented circle bundle over an oriented circle bundle over ... over a circle.

[edit] References

M. Gromov, Almost flat manifolds, J. Differential Geom. 13, 231-241, 1978
E. A. Ruh, Almost flat manifolds, J. Differential Geom. 17, 1-14, 1982

Retrieved from "http://en.wikipedia.org../../../a/l/m/Almost_flat_manifold.html"

Category: Riemannian geometry

Almost flat manifold

From Wikipedia, the free encyclopedia

[edit] References

Views

Navigation

Search