Volume form - Knowlpedia

Let $M$ be an oriented smooth $n$ -manifold. A volume form on $M$ is a smooth nowhere-vanishing differential $n$ -form $\mu$ that is positive on every positively oriented basis of each tangent space.

Riemannian volume form

An oriented Riemannian manifold $(M,g)$ has a canonical volume form $\operatorname{vol}_g$ , characterized in every positively oriented $g$ -orthonormal coframe $(\theta^1,\ldots,\theta^n)$ by

\operatorname{vol}_g=\theta^1\wedge\cdots\wedge\theta^n.

In oriented coordinates, $\operatorname{vol}_g=\sqrt{\det(g_{ij})}\,dx^1\wedge\cdots\wedge dx^n$ . Reversing the orientation changes its sign.

Existence and scope

An $n$ -manifold admits a volume form exactly when it is orientable. A volume form itself chooses an orientation, so no orientation is needed in advance to state the equivalent unoriented version: a smooth nowhere-vanishing top form determines the orientation for which it is positive. A Riemannian metric without an orientation canonically determines a volume density, while a volume form requires a choice of orientation.

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: the chapters on orientations and integration.
John M. Lee, Introduction to Riemannian Manifolds, 2nd ed., Springer, 2018. Publisher record. Relevant: Chapter 2, Riemannian volume.