In mathematics, a smooth compact manifold M is called almost flat there is a Riemannian metric on M-flat, i.e. for the sectional curvature.

-flat metric with diameter 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 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. 526, pp. 21–35, Lecture Notes in Math., 770, Springer, Berlin, 1980.

• Peter Buser and Hermann Karcher. Gromov's almost flat manifolds. Astérisque, 81. Société Mathématique de France, Paris, 1981. 148 pp.

• Peter Buser and Hermann Karcher. The Bieberbach case in Gromov's almost flat manifold theorem. Global differential geometry and global analysis (Berlin, 1979), pp. 82–93, Lecture Notes in Math., 838, Springer, Berlin-New York, 1981.

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