Definition:Euclidean Space/Riemannian Manifold

Definition
Let $\struct {M, g}$ be a Riemannian manifold where $M = \R^n$ for some $n \in \N_{> 0}$ and $g$ is the Euclidean metric.

Then $\struct {M, g}$ is known as the ($n$-dimensional) Euclidean space.