# Definition:Riemannian Metric

## Definition

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 Göttingen.

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.32$: Riemann ($\text {1826}$ – $\text {1866}$)