# 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**.

## Source of Name

This entry was named for Bernhard Riemann.

## Historical Note

The concept of a **Riemannian metric** was originated by Bernhard Riemann in his trial lecture (published as *Ueber die Hypothesen, welche der Geometrie zu Grande liegen*) to apply for position of Privatdozent (unpaid lecturer) at GĂ¶ttingen.

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.32$: Riemann ($\text {1826}$ – $\text {1866}$) - 2018: John M. Lee:
*Introduction to Riemannian Manifolds*(2nd ed.): $\S 2$: Riemannian Metrics. Definitions