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:

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


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