Definition:Euclidean Metric/Riemannian Manifold

Definition
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 $\R^n$:


 * $T_x \R^n \cong \R^n$

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.