# Definition:Riemannian Manifold

## 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 $n$.

## Also see

• Results about Riemannian manifolds can be found here.

## 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 Göttingen.