# Definition:Volume of Compact Riemannian Manifold

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## Definition

Let $\struct {M, g}$ be a Riemannian manifold.

Let $\rd V_g$ be the Riemannian volume form.

Let $M$ be compact.

Then the **volume of the compact Riemannian manifold $M$**, denoted by $\map {\text{Vol}} M$, is defined as:

- $\ds \map {\text{Vol}} M = \int_M \rd V_g$

## Sources

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