Given a Riemannian \(n\)-manifold \((M,g)\). Then we may define the integral of a smooth function \(f\) with compact support by multiplying the Riemannian volume form \(\omega_g\). The resulting product \(f\omega_g\) is a \(n\)-form which we may integrate over \(M\) according to (0x66d5b036) . Therefore the integral of \(f\) over \(M\) is defined by \(\int_{M} f\omega_g\).

Remarks

It is common to write \(dV_g\) instead of \(\omega_g\). However, one should notice that \(dV_g\) is just notation and does not mean that \(\omega_g\) is an exact differential form .

Properties Link to heading

\(f\ge 0\) implies \(\int_{M} fdV_g\ge 0\).
The following formula holds: \begin{equation*} \biggl\lvert \int_{M} f dV_g\biggr\rvert\le \int_{M} \lvert f\rvert dV_g. \end{equation*}