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.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Definitions