# Definition:Riemannian Geometry (Mathematical Branch)

## Definition

**Riemannian geometry** is the branch of differential geometry that studies Riemannian manifolds.

## Also see

- Results about
**Riemannian geometry**can be found here.

## Source of Name

This entry was named for Bernhard Riemann.

## Historical Note

The concept of **Riemannian geometry** originated from 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.

The contents of this lecture proved to be exactly the correct model for Einstein's General Theory of Relativity:

*Riemann's geometry of an $n$-dimensional space bears the same relation to Euclidean geometry of an $n$-dimensional space as the general geometry of curved surfaces bears to the geometry of the plane.*

Hence it has been suggested that this lecture may have been the most important scientific lecture ever given.

## Sources

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