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Categories: Riemannian geometry | Theorems

Almost flat manifold

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

Categories: Riemannian geometry | Theorems

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10-26-2009 08:16:03
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Almost flat manifold