Definition:Riemannian Manifold
Definition
A $n$-dimensional Riemannian manifold is a pair $\struct {M, g}$ where $M$ is a $n$-dimensional smooth manifold and $g$ is a Riemannian metric on $M$.
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Dimension
Let $M$ be a $n$-dimensional Riemannian manifold.
The dimension of $M$ is $n$.
Also known as
A Riemannian manifold is also known as a Riemannian space, or Riemann space.
Some sources use the term Riemannian geometry, although on $\mathsf{Pr} \infty \mathsf{fWiki}$ the term is used to refer to the mathematical branch that studies Riemannian manifolds.
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.
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.32$: Riemann ($\text {1826}$ – $\text {1866}$)
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Riemannian geometry
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Riemannian geometry
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Definitions