Riemannian volume form ($d{\mathrm{vol}}_g$)

On an oriented Riemannian $n$-manifold, the volume form is the unique positive $n$-form taking value one on every positively oriented orthonormal frame. In positive local coordinates,

The total Riemannian volume is the integral of the Riemannian volume form, or its density on a nonorientable manifold. In local coordinates the density is $\sqrt{\det g}d{x}^1\cdots d{x}^d$. It is preserved by Riemannian isometries. On a closed manifold it is the leading coefficient among the integrated heat invariants, and on a surface it is the surface area.

For positive coordinate charts put $J=\partial y/\partial x$. The metric tensor transforms by $g_x=J^T{g}_{y}J$, giving $\det g_x=(\det J{)}^2\det g_y$. The top wedge product of differential forms transforms by $d{y}^1\wedge \cdots \wedge d{y}^n=(\det J)d{x}^1\wedge \cdots \wedge d{x}^n$. Since $\det J>0$, the two factors match. Thus the local expressions define one global Riemannian volume form.

In a positive Riemannian orthonormal frame $X={\sum }_i{f}_i{E}_i$ with dual coframe ${\omega }_i$, the Riemannian volume form is ${\omega }_1\wedge \cdots \wedge {\omega }_n$. Inserting $E_i$ in its first slot omits ${\omega }_i$ and produces sign $(-1{)}^{i-1}$, giving the displayed interior product formula.

At a point in a normal orthonormal frame, torsion-freeness makes the frame brackets vanish there, hence the dual one-forms have zero exterior derivative there. Differentiating the interior product formula then leaves only ${\sum }_i{E}_i(f_i){\omega }_g$, the divergence of a Riemannian vector field. Since the point is arbitrary, the identity holds globally.

On a compact oriented Riemannian manifold without boundary, the Generalized Stokes theorem applied to $f{\iota }_X{\omega }_g$ gives the displayed identity. The product expansion is $d(f{\iota }_X{\omega }_g)=[X(f)+f\mathrm{div}X]{\omega }_g$. With boundary, the omitted boundary integral is ${\int }_{\partial M}f{\iota }_X{\omega }_g$.

The Levi-Civita connection of an oriented Riemannian manifold preserves its Riemannian volume form. In a positively oriented orthonormal frame its connection matrix is skew-symmetric. Differentiating the wedge of the dual coframe gives minus the trace of that matrix times the volume form, which is zero. Equivalently parallel transport preserves the metric and orientation, so it preserves the normalized volume form.

For a nonnegative smooth bump function supported in an open set, multiplying a Riemannian metric by $1+t\eta$ increases volume there for $t>0$ while leaving the metric elsewhere fixed. For distinct regular compact domains a perturbation supported in the interior of one outside the other breaks equal volume, and therefore breaks any isometry between the domains. The perturbations tend to the original metric in the smooth topology.

The $(n-1)$-dimensional volume of the Euclidean unit sphere equals $n$ times the $n$-dimensional volume of its unit ball. If $R={\sum }_i{x}^i{\partial }_i$ is the radial vector field and $\omega =d{x}^1\wedge \cdots \wedge d{x}^n$, then $d({\iota }_R\omega )=n\omega$, and the Generalized Stokes theorem proves the formula.

A differential form is parallel when its covariant derivative vanishes. Parallel transport then preserves it, and every parallel form is harmonic.

The form is preserved by parallel transport; this is the alternating-tensor case of vanishing covariant derivative.

For a function $u$ on a compact Riemannian manifold, its Dirichlet energy is $\frac{1}{2}{\int }_M|\nabla u{|}_g^{2}d{\mathrm{vol}}_g$. Adding a linear source term $-{\int }_{M}fud{\mathrm{vol}}_g$ gives Euler-Lagrange equation $-{\Delta }_{g}u=f$.

For a scalar function $u$ on a Riemannian manifold, the Dirichlet energy is $E(u)=\frac{1}{2}{\int }_M|\nabla u{|}_g^{2}d{\mathrm{vol}}_g$.

For a smooth map between Riemannian manifolds, its Dirichlet energy is half the integral of the squared norm of its differential, using the source and target metrics. When the source is two-dimensional, a conformal change of its metric leaves this energy unchanged. For a J-holomorphic curve into a symplectic manifold with a compatible almost complex structure, the Energy identity for a J-holomorphic curve identifies it with the pulled-back symplectic area. The norm of a differential here is the full tensor norm, summing its squared values on an orthonormal source frame.