Definition:Riemannian Metric

Definition
Let $M$ be a smooth manifold.

Let $p \in M$ be a point in $M$.

Let $T_p M$ be the tangent space of $M$ at $p$ with the inner product $\innerprod \cdot \cdot_p$.

Let $g \in \map {\TT^2} M$ be a smooth covariant 2-tensor field such that for all $p$ its value at $p$ is equal to $\innerprod \cdot \cdot_p$:


 * $\forall p \in M : g_p = \innerprod \cdot \cdot_p$

Then $g$ is known as a Riemannian metric on $M$.

Interpretation
Consider a smooth manifold $\mathcal M$ on the real space $\R^n$.

A Riemannian metric on $\mathcal M$ is a metric $\mathrm d s$ between nearby points $\left({x_1, x_2, \ldots, x_n}\right)$ and $\left({x_1 + \mathrm d x_1, x_2 + \mathrm d x_2, \ldots, x_n + \mathrm d x_n}\right)$ by means of the quadratic differential form:
 * $\displaystyle \mathrm d s^2 = \sum_{i, j \mathop = 1}^n g_{i j} \, \mathrm d x_i \, \mathrm d x_j$

where each $g_{i j}$ is a suitable real-valued function of $x_1, \ldots, x_n$.

Different instances of $g_{i j}$ define different Riemannian geometries on the manifold under discussion.

A manifold with such a Riemannian metric applied is known as a Riemannian manifold.