Almost flat manifold
From Wikipedia, the free encyclopedia
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. On the other hand you can fix the bound of sectional curvature and finally you 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, i.e. a total space of an oriented circle bundle over an oriented circle bundle over ... over a circle.
[edit] References
- M. Gromov, Almost flat manifolds, J. Differential Geom. 13, 231-241, 1978
- E. A. Ruh, Almost flat manifolds, J. Differential Geom. 17, 1-14, 1982
