In mathematics, a smooth

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

manifold ''M'' is called almost flat if for any

\varepsilon>0

there is a

Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space '' ...

g_\varepsilon

on ''M'' such that

\mbox(M,g_\varepsilon)\le 1

and

g_\varepsilon

\varepsilon

-flat, i.e. for the

sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a poi ...

K_

we have

, K_,  < \varepsilon

. Given ''n'', there is a positive number

\varepsilon_n>0

such that if an ''n''-dimensional manifold admits an

\varepsilon_n

-flat metric with diameter

\le 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 In mathematics, a nilmanifold is a differentiable manifold which has a transitive nilpotent group of diffeomorphisms acting on it. As such, a nilmanifold is an example of a homogeneous space and is diffeomorphic to the quotient space N/H, th ...

, which is the total space of a principal torus bundle over a principal torus bundle over a torus.

Notes

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. *. *. Riemannian geometry Manifolds {{differential-geometry-stub