Flat manifold

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°.

A simple example of a flat manifold is given by "n"-dimensional Euclidean space R"n" 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, the 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

:sigma (x, y) = ( cos x, sin x, cos y, sin y ).

Another example of a flat manifold is given by the cylinder

:C = { (x, y, z) in mathbf{R}^{3} | x^{2} + y^{2} = 1, 0 < z < 1 }

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, the universal cover of a complete flat manifold is Euclidean space.

ee also

* Ricci-flat manifold
* Conformally flat manifold

References

External links

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