Almost flat manifold

In mathematics, a smooth compact manifold M is called almost flat if for any

ε

>

0

{\displaystyle \varepsilon >0}

there is a Riemannian metric

g

ε

{\displaystyle g_{\varepsilon }}

on M such that

diam

(

M

,

g

ε

)

≤

1

{\displaystyle {\mbox{diam}}(M,g_{\varepsilon })\leq 1}

and

g

ε

{\displaystyle g_{\varepsilon }}

is

ε

{\displaystyle \varepsilon }

-flat, i.e. for the sectional curvature of

K

g

ε

{\displaystyle K_{g_{\varepsilon }}}

we have

|

K

g

ϵ

|

<

ε

{\displaystyle |K_{g_{\epsilon }}|<\varepsilon }

.

Given

n

{\displaystyle n}

, there is a positive number

ε

n

>

0

{\displaystyle \varepsilon _{n}>0}

such that if an

n

{\displaystyle n}

-dimensional manifold admits an

ε

n

{\displaystyle \varepsilon _{n}}

-flat metric with diameter

≤

1

{\displaystyle \leq 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

{\displaystyle 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.

References

Hermann Karcher. Report on M. Gromov's almost flat manifolds. Séminaire Bourbaki (1978/79), Exp. No.