A flat manifold is a manifold with a Riemannian metric that has zero curvature.
The basic example is Euclidean space with the
usual Riemannian metric 
. On a flat manifold
without boundary, every point has a neighborhood isometric to a neighborhood
in Euclidean space. Consequently, such a manifold
is locally Euclidean in terms of distances and angles,
as well as topologically.
A compact two-dimensional example is the flat torus, realized as a surface
in four-dimensional Euclidean
space. It is the image of 
. By a theorem
of Bieberbach (1911, 1912), every compact connected flat manifold
without boundary has a finite covering
by a flat torus. A compact
flat manifold need not itself be a torus;
the flat klein bottle is a counterexample.
More generally, the universal cover of a connected complete flat manifold
without boundary is Euclidean space.
