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.

-- Albert Einstein

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 ($\text {1826}$ – $\text {1866}$)