Flat manifold

From Wikipedia, the free encyclopedia

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°.

A simple example of a flat manifold is given by n-dimensional Euclidean space Rⁿ with the flat metric g_ij(x) = δ_ij, where δ_ij denotes the Kronecker delta. In fact, any point on a flat manifold has an open neighbourhood that is isometric to an open neighbourhood in Euclidean space. However, that same is not true globally: the 2-dimensional torus T² can be embedded in R⁴ as a flat submanifold via the parametrization σ : T² → R⁴ given by

σ(x,y) = (cosx,sinx,cosy,siny).

Bieberbach's theorem states that all compact flat manifolds are tori. More generally, the universal cover of a complete flat manifold is Euclidean space.

[edit] See also

• Ricci-flat manifold
• Conformally flat manifold

[edit] External links

• Eric W. Weisstein, Flat Manifold at MathWorld.

[edit] References

This article needs additional citations for verification. Please help improve this article by adding reliable references. Unsourced material may be challenged and removed. (August 2007)