Puneet Varma (Editor)

Flat manifold

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

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°.

Contents

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 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

    Flat manifold Wikipedia
    (Text) CC BY-SA