Definition:Flat Riemannian Manifold

Definition
Let $M$ be an $n$-dimensional Riemannian manifold.

$M$ is called the flat Riemannian manifold iff it is locally isometric to the Euclidean space.

That is, for every point $p \in M$ there is a neighborhood that is isometric to an open set in $\R^n$ with the Euclidean metric.