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, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

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

\varepsilon>0

there is a

Riemannian metric In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

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 po ...

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 In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions a ...

\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, 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. *. *. {{Riemannian-geometry-stub Differential geometry Manifolds Riemannian geometry