# Definition:Riemannian Volume Form/Definition 2

Jump to navigation
Jump to search

## Definition

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

Let $T M$ be the tangent bundle of $M$.

Let $\tuple {E_1, \ldots, E_n}$ be a local oriented orthonormal frame of $T M$.

The **Riemannian volume form**, denoted by $\rd V_g$, is an $n$-form such that:

- $\map {\rd V_g} {E_1, \ldots, E_n} = 1$

This article needs to be linked to other articles.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: orientation, formYou 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.): $\S 2$: Riemannian Metrics. Basic Constructions on Riemannian Manifolds