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.

This article, or a section of it, needs explaining. In particular: What does "on the real space $\R^n$" mean? I think it just a bad wording. $\R^n$ is the image of coordinate functions which for each point on the manifold produce $n$ numbers known as the coordinates (c.f. Definition:Chart) You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

This article is complete as far as it goes, but it could do with expansion. In particular: A separate page for $C^k$ Riemannian manifold You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Dimension

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

There are no source works cited for this page. In particular: This definition looks wrong. What does "on $\R^n$" mean? Source citations are highly desirable, and mandatory for all definition pages. Definition pages whose content is wholly or partly unsourced are in danger of having such content deleted. To discuss this page in more detail, feel free to use the talk page.

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