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:

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

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Sources

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