\begin{definition}[Definition:Euclidean Metric/Riemannian Manifold]
Let $x \in \R^n$ be a point.
Let $\tuple {x_1, \ldots, x_n}$ be the standard coordinates.
Let $T_x \R^n$ be the tangent space of $\R^n$ at $x$.
Let $T_x \R^n$ be identified with $\R^n$:
:$T_x \R^n \cong \R^n$
{{Research|why this is needed?}}
Let $v, w \in T_x \R^n$ be vectors such that:
:$\ds v = \sum_{i \mathop = 1}^n v^i \valueat {\partial_i} x$
:$\ds w = \sum_{i \mathop = 1}^n w^i \valueat {\partial_i} x$
Let $g$ be a Riemannian metric such that:
:$\ds g_x = \innerprod v w_x = \sum_{i \mathop = 1}^n v^i w^i$
Then $g$ is called the '''Euclidean metric'''.
{{NamedforDef|Euclid|cat = Euclid}}
\end{definition}