Almost flat manifold

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In mathematics, a smooth compact manifold M is called almost flat if for any \varepsilon>0 there is a Riemannian metric g_\varepsilon on M such that \mbox{diam}(M,g_\varepsilon)\le 1 and g_\varepsilon is \varepsilon-flat, i.e. for sectional curvature of K_{g_\varepsilon} we have |K_{g_\epsilon}| < \varepsilon.

In fact, given n, there is a positive number \varepsilon_n>0 such that if a n-dimensional manifold admits an \varepsilon_n-flat metric with diameter \le 1 then it is almost flat. On the other hand you can fix the bound of sectional curvature and finnally you get the diameter goes 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