# Definition:Volume of Compact Riemannian Manifold

## 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:

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