Volume form

In mathematics, a volume form is a nowhere zero differential "n"-form on an "n"-manifold. Every volume form defines a measure on the manifold, and thus a means to calculate volumes in a generalized sense.

A manifold has a volume form if and only if it is orientable, and orientable manifolds have infinitely many volume forms (details below).There is a generalized notion of pseudo-volume form which exists on any manifold, orientable or not.

Many classes of manifolds come with canonical (pseudo-)volume forms, that is, they have extra structure which allows the choice of a preferred volume form.

In the complex setting, a Kähler manifold with a holomorphic volume form is a Calabi–Yau manifold.

Definition

A volume form is a nowhere vanishing differential form of top degree ("n"-form on an "n"-manifold).

In the language of line bundles, "n"-forms are sections of the line bundle $Omega^n(M)\; =\; Lambda^n(T^*M)$ of top exterior powers, called the determinant line bundle.

For nonorientable manifolds, a volume "pseudo"-form, also called "odd" or "twisted" volume form, may be defined as a nowhere vanishing section of the orientation bundle; this definition also applies for orientable manifolds. In this context (untwisted) differential forms are specified as "even" "n"-forms; unless one is specifically discussing twisted forms, the adjective "even" is omitted for simplicity.

Twisted differential forms were apparently first introduced by de Rham.

Orientation

A manifold has a volume form if and only if it is orientable; this can be taken as a definition of orientability.

In the language of G-structures, a volume form is an SL-structure, As $mbox\{SL\}\; o\; mbox\{GL\}^+$ is a deformation retract(since $mbox\{GL\}^+\; =\; mbox\{SL\}\; imes\; mathbf\{R\}^+$, where the positive reals are embedded as scalar matrices), a manifold admits an SL-structure if and only if it admits a $mbox\{GL\}^+$-structure, which is an orientation.

In the language of line bundles, 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, so again, the existence of a volume form is equivalent to orientability.

For pseudo-volume forms, a pseudo-volume form is an $mbox\{SL\}^pm$-structure, and since $mbox\{SL\}^pm\; o\; mbox\{GL\}$ is a homotopy equivalence (indeed, a deformation retract), every manifold admits a pseudo-volume form. Similarly, the orientation bundle is always trivial, so every manifold admits a pseudo-volume form.

Relation to measures

Any manifold admits a volume pseudo-form, as the orientation bundle is trivial (as a line bundle). Given a volume form ω on an oriented manifolds, the density |ω| is a volume pseudo-form on the nonoriented manifold obtained by forgetting the orientation.

Any volume pseudo-form ω (and therefore also any volume form) defines a measure on the Borel sets by:$mu\_omega(U)=int\_Uomega.\; ,!$

The difference is that while a measure can be integrated over a (Borel) "subset", a volume form can only be integrated over an "oriented" cell. In single variable calculus, writing $int\_b^a\; f,dx\; =\; -int\_a^b\; f,dx$ considers $dx$ as a volume form, not simply a measure, and $int\_b^a$ 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, their Radon–Nikodym derivative with respect to a given volume form needn't be absolutely continuous.

Examples

Lie groups

For any Lie group, a natural volume form may be defined by translation. That is, if ω_"e" is an element of , then a left-invariant form may be defined by $omega\_g=L\_g^*omega\_e$, 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 the Haar measure.

Symplectic manifolds

Any symplectic manifold (or indeed any almost symplectic manifold) has a natural volume form. If "M" is a 2"n"-dimensional manifold with symplectic form ω, then ω^"n" is nowhere zero as a consequence of the nondegeneracy of 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 is Kähler.

Riemannian volume form

Any Riemannian (or pseudo-Riemannian) manifold has a natural volume (or pseudo volume) form. In local coordinates, it can be expressed as:$omega\; =\; sqrt\; dx^1wedge\; dots\; wedge\; dx^n$where the manifold is an "n" dimensional manifold. Here, $|g|$ is the absolute value of the determinant of the metric tensor on the manifold. The $dx^i$ are the 1-forms providing a basis for the cotangent bundle of the manifold.

A number of different notations are in use for the volume form. These include

:$omega\; =\; mathrm\{vol\}\_n\; =\; epsilon\; =\; *(1)\; .\; ,!$

Here, the ∗ is the Hodge dual, thus the last form, ∗(1), emphasizes that the volume form is the Hodge dual of the constant map on the manifold.

Although the Greek letter ω is frequently used to denote the volume form, this notation is hardly universal; the symbol ω often carries many other meanings in differential geometry (such as a symplectic form); thus, the appearance of ω in a formula does not necessarily mean that it is the volume form.

Volume form of a surface

A simple example of a volume form can be explored by considering a two-dimensional surface embedded in "n"-dimensional Euclidean space. Consider a subset $U\; subset\; mathbf\{R\}^2$ and a mapping function

:$phi:U\; o\; mathbf\{R\}^n$

thus defining a surface embedded in $mathbf\{R\}^n$. The Jacobian matrix of the mapping is

:$lambda\_\{ij\}=frac\{partial\; phi\_i\}\; \{partial\; u\_j\}$

with index "i" running from 1 to "n", and "j" running from 1 to 2. The Euclidean metric in the "n"-dimensional space induces a metric $g=lambda^Tlambda$ on the set "U", with matrix elements

:$g\_\{ij\}=sum\_\{k=1\}^n\; lambda\_\{ki\}\; lambda\_\{kj\}=\; sum\_\{k=1\}^nfrac\{partial\; phi\_k\}\; \{partial\; u\_i\}frac\{partial\; phi\_k\}\; \{partial\; u\_j\}$

The determinant of the metric is given by

:$det\; g\; =\; left|\; frac\{partial\; phi\}\; \{partial\; u\_1\}\; wedgefrac\{partial\; phi\}\; \{partial\; u\_2\}\; ight|^2\; =\; det\; (lambda^T\; lambda)$

where $wedge$ is the wedge product. For a regular surface, this determinant is non-vanishing; equivalently, the Jacobian matrix has rank 2.

Now consider a change of coordinates on "U", given by a diffeomorphism

:$f\; colon\; U\; o\; U\; ,\; ,!$ so that the coordinates $(u\_1,u\_2)$ are given in terms of $(v\_1,v\_2)$ by $(u\_1,u\_2)=\; f(v\_1,v\_2)$. The Jacobian matrix of this transformation is given by

:$F\_\{ij\}=\; frac\{partial\; f\_i\}\; \{partial\; v\_j\}$

In the new coordinates, we have

:$frac\{partial\; phi\_i\}\; \{partial\; v\_j\}\; =\; sum\_\{k=1\}^2\; frac\{partial\; phi\_i\}\; \{partial\; u\_k\}frac\{partial\; f\_k\}\; \{partial\; v\_j\}$

and so the metric transforms as

:$ilde\{g\}\; =\; F^T\; g\; F$

where $ilde\{g\}$ is the metric in the "v" coordinate system. The determinant is

:$det\; ilde\{g\}\; =\; det\; g\; (det\; F)^2$.

Given the above construction, it should now be straightforward to understand how the volume form is invariant under a change of coordinates. In two dimensions, the volume is just the area. The area of a subset $Bsubset\; U$ is given by the integral

:

Thus, in either coordinate system, the volume form takes the same expression: the expression of the volume form is invariant under a change of coordinates.

Note that there was nothing particular to two dimensions in the above presentation; the above trivially generalizes to arbitrary dimensions.

Invariants of a volume form

Volume forms are not unique; they form a torsor over non-vanishing functions on the manifold, as follows. This is a geometric form of the Radon–Nikodym theorem.

Given a non-vanishing function "f" on "M", and a volume form $omega$, $fomega$ is a volume form on "M". Conversely, given two volume forms $omega,\; omega\text{'}$, 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 times Lebesgue measure, and their ratio is the ratio of the functions, which is independent of choice of coordinates. Intrinsically, it is the Radon–Nikodym derivative of $omega\text{'}$ with respect to $omega$.

No local structure

A volume form has no local structure: any two volume forms (on manifolds of the same dimension) are locally isomorphic.

Formally, this statement means that given two manifolds of the same dimension $M,N$ with volume forms $omega\_M,\; omega\_N$, for any points $min\; M,\; nin\; N$, there is a map $fcolon\; U\; o\; V$ (where $U$ is a neighborhood of $m$ and $V$ is a neighborhood of $n$) such that the volume form on $N$ (restricted to the neighborhood $V$) pulls back to volume form on $M$ (restricted to the neighborhood $U$): $f^*omega\_Nvert\_V\; =\; omega\_Mvert\_U$.Differentiable manifolds of a given dimension are locally diffeomorphic; the added criterion is that the volume form pulls back to the volume form.

In one dimension, one can prove it thus:given a volume form $omega$ on $mathbf\{R\}$, define:$f(x)\; :=\; int\_0^x\; omega$Then the standard Lebesgue measure $dx$ pulls back to $omega$ 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 $mathbf\{R\}\; imesmathbf\{R\}^\{n-1\}$, 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 $mathbf\{R\}^n$. On a disconnected manifold, the volume of each connected component is the invariant.

In symbol, if $fcolon\; M\; o\; N$ 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 under covering 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 $mathbf\{R\}\; o\; S^1$), a volume form on a finite volume manifold pulls back to a volume form on an infinite volume manifold.

ee also

* Poincaré metric provides a review of the volume form on the complex plane.
* measure (mathematics)

References

* Michael Spivak, "Calculus on Manifolds", (1965) W.A. Benjamin, Inc. Reading, Massachusetts ISBN 0-8053-9021-9 "(Provides an elementary introduction to the modern notation of differential geometry, assuming only a general calculus background)"