Almost flat manifold Guide, Meaning , Facts, Information and Description

In mathematics, a smooth compact manifold M is called almost flat if for any there is a Riemannian metric on M such that and

is -flat, i.e. for sectional curvature of  we have .

In fact, given n, there is a positive number such that if a n-dimensional manifold admits an -flat metric with diameter 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.

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