Volume form

2020 Mathematics Subject Classification: Primary: 53B20 [MSN][ZBL] volume element.

The volume form is a special differential form defined on oriented Riemannian manifolds and which introduces a natural concept of measure on the manifold.

On vector spaces

First consider a real vector space $V$ of dimension $n$ with a given orientation and an inner product. The corresponding volume form, or volume element, is the unique element $\omega$ of the space $\Lambda^n V$ of $n$-forms (namely alternating multilinear maps $L : V^n \to \mathbb R$; cf. Exterior form), such that $\omega (v_1, \ldots, v_n) =1$ for each orthonormal (with respect to the given inner product) positively oriented basis of $V$. Recall that $\Lambda^n V$ is one-dimensional. If $V = \mathbb R^n$ with the standard inner product and orientation, then, for each $n$-tuple of vectors $w_1, \ldots, w_n \in \mathbb R^n$, $\omega (w_1, \ldots, w_n)$ is the determinant of the $n\times n$ matrix whose $ij$ entry is the $i$-th corrdinate (in the standared basis) of $w_j$. Moreover, $|\omega (v_1, \ldots, v_n)|$ is the Lebesgue measure (i.e. the $n$-dimensional volume) of the parallelopipedon $P$ spanned by the vectors $v_j$, namely \[ P = \left\{\sum_j \lambda_j v_j : \lambda_j\in [0,1]\right\}\, . \] The latter property gives an intuitive justification for the terms volume form and volume element: in particular, if $U$ is an open bounded subset of $\mathbb R^n$, then \begin{equation}\label{e:integra} \int_U \omega \end{equation} is the Lebesgue measure of $U$ (for the definition of the integral in \eqref{e:integra} see Integration on manifolds).

On Riemannian manifolds

If $(\Sigma, g)$ is an oriented Riemannian manifold, then the volume form $\omega$ on $\Sigma$ is the unique element of $\Lambda^n M$ such that, for each $x\in M$, $\omega|_x$ is the volume form on the vector space $T_x M$ relative to the inner product $g_x$ and the orientation chosen. One often writes ${\rm dV}$ or ${\rm dvol}$, for the volume form on $\Sigma$, even though the volume form might not be exact (indeed the volume form is never exact if $M$ is compact and without boundary).

Formula in local coordinates

Consider a local coordinate patch $U$ on $\Sigma$. Let $x_1, \ldots, x_n$ be the corresponding coordinates and denote by $g_{ij} (x)$ the inner product (at the point $x$) of the vector fields $\frac{\partial}{\partial x_i}$ and $\frac{\partial}{\partial x_j}$(see Tangent vector), namely \[ g_{ij} (x) = g_x \left(\frac{\partial}{\partial x_i}, \frac{\partial}{\partial x_j}\right)\, . \] With a slight abuse of notation, let $\det g (x)$ be the determinant of the matrix $(g_{ij} (x))_{ij}$. We then have the following formula for the volume form ${\rm dvol}$ on $U$: \[ {\rm dvol} = \varepsilon\, \sqrt{\det g (x)}\, dx_1\wedge \ldots \wedge dx_n \] where $\varepsilon =1 $ if $\frac{\partial}{\partial x_1}, \ldots, \frac{\partial}{\partial x_n}$ is positively oriented, and $-1$ otherwise.

Volume measure

The volume form defines naturally a measure $\mu$ on any Riemannian manifold. Indeed, given a smooth continuous compactly supported function $f$ we can define \[ \int_\Sigma f d\mu \] as the integral of the form $f {\rm dvol}$, in the sense of integration on manifolds. It can be checked that such integral coincides with the Lebesgue integral with respect to a $\sigma$-additive measure on an appropriate $\sigma$-algebra (assuming that the manifold is second countable).

Divergence and Hodge star

Assume $(\Sigma, g)$ is an oriented Riemannian manifold. If $\nabla$ is the Levi-Civita connection on $\Sigma$, the divergence of a vector field $X$ is then given, at the point $x$, by \[ {\rm div}\, X = \sum_j \nabla_{e_j} X\, , \] where $e_1, \ldots, e_n$ is any orthonormal frame defined in a neighborhood of $x$. In local coordinates, if $X = \sum \psi^j \frac{\partial}{\partial x_j}$, then \[ {\rm div}\, X = (\det g)^{-1/2} \sum_j \frac{\partial}{\partial x_j} \left( (\det g)^{1/2} \psi^j\right)\, . \] Let $*$ denote the Hodge star operator on the oriented Riemannian manifold $\Sigma$ (cf. Laplace operator).It then turns out that \begin{equation}\label{e:star} d (*X) = {\rm div}\, X\, {\rm dvol}\, . \end{equation}

Stokes and divergence theorems

The latter identity gives a link between Stokes theorem on differential forms on the divergence theorem (also called Gauss-Green theorem or Ostrogradski formula). More precisely, assume that $(\Sigma,g)$ is a compact oriented Riemannian manifold with boundary $\partial \Sigma$. Denote by:

• ${\rm d Vol}$ the volume form on $\Sigma$;
• ${\rm dvol}$ the volume form on $\partial \Sigma$;
• $\nu$ the exterior unit normal to $\partial \Sigma$ (namely, if $e_1, \ldots, e_{n-1}$ is a positively oriented ortonormal frame on $U\cap \partial \Sigma$, then $e_1, \ldots, e_{n-1}, \nu$ is a positively oriented orthonormal frame in $U\cap \Sigma$).

Stokes theorem together with \eqref{e:star} implies that \[ \int_\Sigma {\rm div}\, X\, {\rm dVol} = \int_{\partial \Sigma} *X\, . \] On the other it can be shown that \[ \int_{\partial \Sigma} *X = \int_{\partial \Sigma} g (X, \nu)\, {\rm dvol}\, , \] thereby concluding \[ \int_\Sigma {\rm div}\, X\, {\rm dVol} = \int_{\partial \Sigma} g (X, \nu)\, {\rm dvol}\, . \]

Area formula

For $C^1$ submanifolds of the Euclidean space $\mathbb R^N$, the volume form can be related to a measure-theoretic concept of volume. More precisely, assume $\Sigma$ is a (compact) $m$-dimensional $C^1$ submanifold of $\mathbb R^N$. Let ${\rm dvol}$ be the volume form on $\Sigma$ related to the Riemannian structure given by the restriction of the standard Euclidean inner product on the tangent space to $\Sigma$. The area formula implies that, if we denote by $\mathcal{H}^m$ the $m$-dimensional Hausdorff measure on $\mathbb R^N$, then \[ \int_\Sigma {\rm dvol} = \mathcal{H}^m (\Sigma)\, . \]

References

[Be]	Y. Choquet-Bruhat, C. DeWitt-Morette, M. Dillard-Bleick, "Analysis, manifolds, and physics" , North-Holland (1977) (Translated from French)
[Ha]	M. Hazewinkel, "A tutorial introduction to differentiable manifolds and calculus on manifolds" W. Schiehlen (ed.) W. Wedig (ed.) , Analysis and estimation of stochastic mechanical systems , Springer (Wien) (1988) pp. 316–340
[Sp]	M. Spivak, "Calculus on manifolds" , Benjamin (1965)