\begin{definition}[Definition:Flat Riemannian Manifold]
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.
\end{definition}