A flat manifold is a manifold with a Riemannian metric that has zero curvature. The basic example is Euclidean
space with the usual metric 
. 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 
.
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.
