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.

Dimension

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

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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 ($\text {1826}$ – $\text {1866}$)
• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Definitions