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 $\mathbb{R}^3$).

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 C¹ map (by the Nash embedding theorem) but not with a C² 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.

See also

• Space forms
• Crystallographic groups
• Ricci-flat manifold
• Conformally flat manifold
• Affine manifold

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), "Crystallographic group", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

External links

• Weisstein, Eric W., "Flat Manifold", MathWorld.