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.

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, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

External links

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

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