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°. 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 R 3 {displaystyle mathbb {R} ^{3}} ). 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. For the complete list of the 6 orientable and 4 non-orientable compact examples see Seifert fiber space.