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

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 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.