Flat Manifold

A manifold with a Riemannian metric that has zero curvature is a flat manifold. The basic example is Euclidean space with the usual metric ds^2=sum_(i)dx_i^2. In fact, any point on a flat manifold has a neighborhood isometric to a neighborhood in Euclidean space. A flat manifold is locally Euclidean in terms of distances and angles, as well as merely topologically locally Euclidean, as all manifolds are.

The simplest nontrivial examples occur as surfaces in four dimensional space. For instance, the flat torus is a flat manifold. It is the image of f(x,y)=(cosx,sinx,cosy,siny). A theorem due to Bieberbach says that all compact flat manifolds are tori. More generally, the universal cover of a complete flat manifold is Euclidean space.

See also

Curvature, Exponential Map, Flat, Isometry, Torus, Universal Cover

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Flat Manifold." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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