# Definition:Riemannian Manifold

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## Definition

A **Riemannian manifold** is a smooth manifold on the real space $\R^n$ upon which a Riemannian metric has been imposed.

### Dimension

The **dimension** of the **Riemannian manifold** on $\R^n$ is $n$.

## Source of Name

This entry was named for Bernhard Riemann.

## Historical Note

The concept of a **Riemannian manifold** 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