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).
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Examples
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
References
- Bieberbach, L. (1911), "Über die Bewegungsgruppen der Euklidischen Räume I", Mathematische Annalen 70 (3): 297–336, doi:.
- Bieberbach, L. (1912), "Über die Bewegungsgruppen der Euklidischen Räume II: Die Gruppen mit einem endlichen Fundamentalbereich", Mathematische Annalen 72 (3): 400–412, doi:.
- Schoenflies, A. (1891), Kristallsysteme und Kristallstruktur, Teubner.
- Vinberg, E.B. (2001), "Crystallographic group", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
External links
- Weisstein, Eric W., "Flat Manifold" from MathWorld.
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