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).
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 ).
- The line
- The circle
- 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.
For the complete list of the 6 orientable and 4 non-orientable compact examples see Seifert fiber space.
- Euclidean space
- Products of flat manifolds
- Quotients of flat manifolds by groups acting freely.
Relation to amenability
This is a consequence of the Adams-Ballmann theorem (1998), 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.
- 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.
- 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.