In mathematics, a smooth compact manifold M is called almost flat if for any
there is a Riemannian metric
diam
-flat, i.e. for the sectional curvature of
, there is a positive number
-dimensional manifold admits an
-flat metric with diameter
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.
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.
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