Volume form

Inmathematics,avolume formortop-dimensional formis adifferential formof degree equal to thedifferentiable manifolddimension. Thus on a manifold$M$of dimension$n$,a volume form is an$n$-form. It is an element of the space ofsectionsof theline bundle$\textstyle {\bigwedge }^{n}(T^{*}M)$,denoted as$\Omega ^{n}(M)$.A manifold admits a nowhere-vanishing volume form if and only if it is orientable. Anorientable manifoldhas infinitely many volume forms, since multiplying a volume form by a nowhere-vanishing real valued function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of adensity.

A volume form provides a means to define theintegralof afunctionon a differentiable manifold. In other words, a volume form gives rise to ameasurewith respect to which functions can be integrated by the appropriateLebesgue integral.The absolute value of a volume form is avolume element,which is also known variously as atwisted volume formorpseudo-volume form.It also defines a measure, but exists on any differentiable manifold, orientable or not.

Kähler manifolds,beingcomplex manifolds,are naturally oriented, and so possess a volume form. More generally, the$n$thexterior powerof the symplectic form on asymplectic manifoldis a volume form. Many classes of manifolds have canonical volume forms: they have extra structure which allows the choice of a preferred volume form. Orientedpseudo-Riemannian manifoldshave an associated canonical volume form.

Orientation

The following will only be about orientability ofdifferentiablemanifolds (it's a more general notion defined on any topological manifold).

A manifold isorientableif it has acoordinate atlasall of whose transition functions have positiveJacobian determinants.A selection of a maximal such atlas is an orientation on$M.$A volume form$\ Omega$on$M$gives rise to an orientation in a natural way as the atlas of coordinate charts on$M$that send$\ Omega$to a positive multiple of the Euclidean volume form$dx^{1}\wedge \cdots \wedge dx^{n}.$

A volume form also allows for the specification of a preferred class offrameson$M.$Call a basis of tangent vectors$(X_{1},\ldots,X_{n})$right-handed if $$\ Omega \left(X_{1},X_{2},\ldots,X_{n}\right)>0.$$

The collection of all right-handed frames isacted uponby thegroup$\mathrm {GL} ^{+}(n)$ofgeneral linearmappings in$n$dimensions with positive determinant. They form aprincipal$\mathrm {GL} ^{+}(n)$sub-bundleof thelinear frame bundleof$M,$and so the orientation associated to a volume form gives a canonical reduction of the frame bundle of$M$to a sub-bundle with structure group$\mathrm {GL} ^{+}(n).$That is to say that a volume form gives rise to$\mathrm {GL} ^{+}(n)$-structureon$M.$More reduction is clearly possible by considering frames that have

$\ Omega \left(X_{1},X_{2},\ldots,X_{n}\right)=1.$(1)

Thus a volume form gives rise to an$\mathrm {SL} (n)$-structure as well. Conversely, given an$\mathrm {SL} (n)$-structure, one can recover a volume form by imposing (1) for the special linear frames and then solving for the required$n$-form$\ Omega$by requiring homogeneity in its arguments.

A manifold is orientable if and only if it has a nowhere-vanishing volume form. Indeed,$\mathrm {SL} (n)\to \mathrm {GL} ^{+}(n)$is adeformation retractsince$\mathrm {GL} ^{+}=\mathrm {SL} \times \mathbb {R} ^{+},$where thepositive realsare embedded as scalar matrices. Thus every$\mathrm {GL} ^{+}(n)$-structure is reducible to an$\mathrm {SL} (n)$-structure, and$\mathrm {GL} ^{+}(n)$-structures coincide with orientations on$M.$More concretely, triviality of the determinant bundle$\Omega ^{n}(M)$is equivalent to orientability, and a line bundle is trivial if and only if it has a nowhere-vanishing section. Thus, the existence of a volume form is equivalent to orientability.

Relation to measures

See also:Density on a manifold

Given a volume form$\ Omega$on an oriented manifold, thedensity$|\ Omega |$is a volumepseudo-formon the nonoriented manifold obtained by forgetting the orientation. Densities may also be defined more generally on non-orientable manifolds.

Any volume pseudo-form$\ Omega$(and therefore also any volume form) defines a measure on theBorel setsby $$\mu _{\ Omega }(U)=\int _{U}\ Omega.$$

The difference is that while a measure can be integrated over a (Borel)subset,a volume form can only be integrated over anorientedcell. In single variablecalculus,writing${\textstyle \int _{b}^{a}f\,dx=-\int _{a}^{b}f\,dx}$considers$dx$as a volume form, not simply a measure, and${\textstyle \int _{b}^{a}}$indicates "integrate over the cell$[a,b]$with the opposite orientation, sometimes denoted${\overline {[a,b]}}$".

Further, general measures need not be continuous or smooth: they need not be defined by a volume form, or more formally, theirRadon–Nikodym derivativewith respect to a given volume form need not beabsolutely continuous.

Divergence

Given a volume form$\ Omega$on$M,$one can define thedivergenceof avector field$X$as the unique scalar-valued function, denoted by$\operatorname {div} X,$satisfying $$(\operatorname {div} X)\ Omega =L_{X}\ Omega =d(X\mathbin {\!\rfloor } \ Omega ),$$ where$L_{X}$denotes theLie derivativealong$X$and$X\mathbin {\!\rfloor } \ Omega$denotes theinterior productor the leftcontractionof$\ Omega$along$X.$If$X$is acompactly supportedvector field and$M$is amanifold with boundary,thenStokes' theoremimplies $$\int _{M}(\operatorname {div} X)\ Omega =\int _{\partial M}X\mathbin {\!\rfloor } \ Omega,$$ which is a generalization of thedivergence theorem.

Thesolenoidalvector fields are those with$\operatorname {div} X=0.$It follows from the definition of the Lie derivative that the volume form is preserved under theflowof a solenoidal vector field. Thus solenoidal vector fields are precisely those that have volume-preserving flows. This fact is well-known, for instance, influid mechanicswhere the divergence of a velocity field measures the compressibility of a fluid, which in turn represents the extent to which volume is preserved along flows of the fluid.

Special cases

Lie groups

For anyLie group,a natural volume form may be defined by translation. That is, if$\ Omega _{e}$is an element of${\textstyle \bigwedge }^{n}T_{e}^{*}G,$then a left-invariant form may be defined by$\ Omega _{g}=L_{g^{-1}}^{*}\ Omega _{e},$where$L_{g}$is left-translation. As a corollary, every Lie group is orientable. This volume form is unique up to a scalar, and the corresponding measure is known as theHaar measure.

Symplectic manifolds

Anysymplectic manifold(or indeed anyalmost symplectic manifold) has a natural volume form. If$M$is a$2n$-dimensional manifold withsymplectic form$\ Omega,$then$\ Omega ^{n}$is nowhere zero as a consequence of thenondegeneracyof the symplectic form. As a corollary, any symplectic manifold is orientable (indeed, oriented). If the manifold is both symplectic and Riemannian, then the two volume forms agree if the manifold isKähler.

Riemannian volume form

Anyorientedpseudo-Riemannian(includingRiemannian)manifoldhas a natural volume form. Inlocal coordinates,it can be expressed as $$\ Omega ={\sqrt {|g|}}dx^{1}\wedge \dots \wedge dx^{n}$$ where the$dx^{i}$are1-formsthat form a positively oriented basis for thecotangent bundleof the manifold. Here,$|g|$is the absolute value of thedeterminantof the matrix representation of themetric tensoron the manifold.

The volume form is denoted variously by $$\ Omega =\mathrm {vol} _{n}=\varepsilon ={\star }(1).$$

Here, the${\star }$is theHodge star,thus the last form,${\star }(1),$emphasizes that the volume form is the Hodge dual of the constant map on the manifold, which equals theLevi-Civitatensor$\varepsilon.$

Although the Greek letter$\ Omega$is frequently used to denote the volume form, this notation is not universal; the symbol$\ Omega$often carries many other meanings indifferential geometry(such as a symplectic form).

Invariants of a volume form

Volume forms are not unique; they form atorsorover non-vanishing functions on the manifold, as follows. Given a non-vanishing function$f$on$M,$and a volume form$\ Omega,$$f\ Omega$is a volume form on$M.$Conversely, given two volume forms$\ Omega,\ Omega ',$their ratio is a non-vanishing function (positive if they define the same orientation, negative if they define opposite orientations).

In coordinates, they are both simply a non-zero function timesLebesgue measure,and their ratio is the ratio of the functions, which is independent of choice of coordinates. Intrinsically, it is theRadon–Nikodym derivativeof$\ Omega '$with respect to$\ Omega.$On an oriented manifold, the proportionality of any two volume forms can be thought of as a geometric form of theRadon–Nikodym theorem.

No local structure

A volume form on a manifold has no local structure in the sense that it is not possible on small open sets to distinguish between the given volume form and the volume form on Euclidean space (Kobayashi 1972). That is, for every point$p$in$M,$there is an open neighborhood$U$of$p$and adiffeomorphism$\varphi$of$U$onto an open set in$\mathbb {R} ^{n}$such that the volume form on$U$is thepullbackof$dx^{1}\wedge \cdots \wedge dx^{n}$along$\varphi.$

As a corollary, if$M$and$N$are two manifolds, each with volume forms$\ Omega _{M},\ Omega _{N},$then for any points$m\in M,n\in N,$there are open neighborhoods$U$of$m$and$V$of$n$and a map$f:U\to V$such that the volume form on$N$restricted to the neighborhood$V$pulls back to volume form on$M$restricted to the neighborhood$U$:$f^{*}\ Omega _{N}\vert _{V}=\ Omega _{M}\vert _{U}.$

In one dimension, one can prove it thus: given a volume form$\ Omega$on$\mathbb {R},$define $$f(x):=\int _{0}^{x}\ Omega.$$ Then the standardLebesgue measure$dx$pulls backto$\ Omega$under$f$:$\ Omega =f^{*}dx.$Concretely,$\ Omega =f'\,dx.$In higher dimensions, given any point$m\in M,$it has a neighborhood locally homeomorphic to$\mathbb {R} \times \mathbb {R} ^{n-1},$and one can apply the same procedure.

Global structure: volume

A volume form on a connected manifold$M$has a single global invariant, namely the (overall) volume, denoted$\mu (M),$which is invariant under volume-form preserving maps; this may be infinite, such as for Lebesgue measure on$\mathbb {R} ^{n}.$On a disconnected manifold, the volume of each connected component is the invariant.

In symbols, if$f:M\to N$is a homeomorphism of manifolds that pulls back$\ Omega _{N}$to$\ Omega _{M},$then $$\mu (N)=\int _{N}\ Omega _{N}=\int _{f(M)}\ Omega _{N}=\int _{M}f^{*}\ Omega _{N}=\int _{M}\ Omega _{M}=\mu (M)\,$$ and the manifolds have the same volume.

Volume forms can also be pulled back undercovering maps,in which case they multiply volume by the cardinality of the fiber (formally, by integration along the fiber). In the case of an infinite sheeted cover (such as$\mathbb {R} \to S^{1}$), a volume form on a finite volume manifold pulls back to a volume form on an infinite volume manifold.

See also

• Cylindrical coordinate system § Line and volume elements
• Measure (mathematics)– Generalization of mass, length, area and volume
• Poincaré metricprovides a review of the volume form on thecomplex plane
• Spherical coordinate system § Integration and differentiation in spherical coordinates

References

• Kobayashi, S. (1972),Transformation Groups in Differential Geometry,Classics in Mathematics, Springer,ISBN3-540-58659-8,OCLC31374337.
• Spivak, Michael(1965),Calculus on Manifolds,Reading, Massachusetts: W.A. Benjamin, Inc.,ISBN0-8053-9021-9.