Definition:Riemannian Density

Definition

Let $\struct {M, g}$ be an $n$-dimensional Riemannian manifold.

Let $\tuple {E_i}$ be an local orthonormal frame.

Let $\mu$ be a positive smooth density such that:

$\forall \tuple {E_i} : \map \mu {E_1, \ldots, E_n} = 1$

Then $\mu$ is known as the Riemannian density.

This article needs to be linked to other articles. In particular: tensor density You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links. 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 {{MissingLinks}} from the code.

Further research is required in order to fill out the details. In particular: "for all frames" You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by finding out more. 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 {{Research}} from the code.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Basic Constructions on Riemannian Manifolds