# Flat manifold

In mathematics, a Riemannian manifold is said to be **flat** if its curvature is everywhere zero. Intuitively, a flat manifold is one that "locally looks like" Euclidean space in terms of distances and angles, e.g. the interior angles of a triangle add up to 180°.

The universal cover of a complete flat manifold is Euclidean space. This can be used to prove the theorem of Bieberbach (1911, 1912) that all compact flat manifolds are finitely covered by tori; the 3-dimensional case was proved earlier by Schoenflies (1891).

## Examples

The following manifolds can be endowed with a flat metric. Note that this may not be their 'standard' metric (for example, the flat metric on the 2-dimensional torus is not the metric induced by its usual embedding into ).

### Dimension 1

- The line
- The circle

### Dimension 2

- The plane
- The cylinder
- The Moebius band
- The Klein bottle
- The 2-dimensional torus. A flat torus can be isometrically embedded in three-dimensional Euclidean space with a
*C*^{1}map (by the Nash embedding theorem) but not with a*C*^{2}map, and the Clifford torus provides an isometric analytic embedding of a flat torus in four dimensions.

There are 17 compact 2-dimensional orbifolds with flat metric (including the torus and Klein bottle), listed in the article on orbifolds, that correspond to the 17 wallpaper groups.

### Dimension 3

For the complete list of the 6 orientable and 4 non-orientable compact examples see Seifert fiber space.

### Higher dimensions

- Euclidean space
- Tori
- Products of flat manifolds
- Quotients of flat manifolds by groups acting freely.

## Relation to amenability

Among all closed manifolds with non-positive sectional curvature, flat manifolds are characterized as precisely those with an amenable fundamental group.

This is a consequence of the Adams-Ballmann theorem (1998),[1] which establishes this characterization in the much more general setting of discrete cocompact groups of isometries of Hadamard spaces. This provides a far-reaching generalisation of Bieberbach's theorem.

The discreteness assumption is essential in the Adams-Ballmann theorem: otherwise, the classification must include symmetric spaces, Bruhat-Tits buildings and Bass-Serre trees in view of the "indiscrete" Bieberbach theorem of Caprace-Monod.[2]

## See also

## References

- Bieberbach, L. (1911), "Über die Bewegungsgruppen der Euklidischen Räume I",
*Mathematische Annalen*,**70**(3): 297–336, doi:10.1007/BF01564500.

- Bieberbach, L. (1912), "Über die Bewegungsgruppen der Euklidischen Räume II: Die Gruppen mit einem endlichen Fundamentalbereich",
*Mathematische Annalen*,**72**(3): 400–412, doi:10.1007/BF01456724.

- Schoenflies, A. (1891),
*Kristallsysteme und Kristallstruktur*, Teubner.

- Vinberg, E.B. (2001) [1994], "Crystallographic group", in Hazewinkel, Michiel (ed.),
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

## External links

## References

- Adams, S.; Ballmann, W. (1998). "Amenable isometry groups of Hadamard spaces".
*Math. Ann*.**312**(1): 183–195. - Caprace, P.-E.; Monod, N. (2015). "An indiscrete Bieberbach theorem: from amenable CAT(0) groups to Tits buildings".
*J. Ecole Polytechnique*.**2**: 333–383.