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
diam
(
M
,
g
ε
)
≤
1
and
g
ε
is
ε
-flat, i.e. for the sectional curvature of
K
g
ε
we have
|
K
g
ϵ
|
<
ε
.
Given n, there is a positive number
ε
n
>
0
such that if an n-dimensional manifold admits an
ε
n
-flat metric with diameter
≤
1
then it is almost flat. 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.
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, which is the total space of a principal torus bundle over a principal torus bundle over a torus.
