- Flat manifold
In
mathematics , aRiemannian manifold is said to be flat if itscurvature 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"-
dimension al Euclidean space R"n" with the flat metric "g""ij"("x") = "δ""ij", where "δ""ij" denotes theKronecker delta . In fact, any point on a flat manifold has anopen neighbourhood that is isometric to an open neighbourhood in Euclidean space. However, the same is not true globally: the 2-dimensionaltorus T2 can be embedded in R4 as a flat submanifold via the parametrization "σ" : T2 → R4 given by:
Another example of a flat manifold is given by the
cylinder :
with its usual Riemannian structure. On the other hand, the
unit sphere with its usual Riemannian structure is not a flat manifold: it has constant, everywhere positive curvature.Bieberbach's theorem states that all compact flat manifolds are tori. More generally, theuniversal cover of a complete flat manifold is Euclidean space.ee also
*
Ricci-flat manifold
*Conformally flat manifold References
External links
*
Wikimedia Foundation. 2010.