Flat manifold

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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 ).

Dimension 1[edit]

  • The line
  • The circle

Dimension 2[edit]

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[edit]

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

Higher dimensions[edit]

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

See also[edit]


  • 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.

External links[edit]