volume form (changes) in nLab

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Idea

The volume form on a finite-dimensional oriented (pseudo)-Riemannian manifold (X,g)(X,g) is the differential form whose integral over pieces of XX computes the volume of XX as measured by the metric gg.

If the manifold is unoriented, then we get a volume pseudoform instead, or equivalently a volume density (of weight 11). We can also consider volume (pseudo)-forms in the absence of a metric, in which case we have a choice of volume forms.

Definition

General

For XX a general smooth manifold of finite dimension nn, a volume form on XX is a nondegenerate (nowhere vanishing) differential nn-form on XX, equivalently a nondegenerate section of the canonical line bundle on XX. A volume pseudoform or volume element on XX is a positive definite density (of rank 11) on XX, or equivalently a positive definite differential nn-pseudoform on XX.

A volume form defines an orientation on XX, the one relative to which it is positive definite. If XX is already oriented, then we require the orientations to agree (to have a volume form on XX qua oriented manifold); that is, a volume form on an oriented manifold must be positive definite (just as a volume pseudoform on any manifold must be). In this situation, there is essentially no difference between a form and a pseudoform, hence no difference between a volume form and a volume pseudoform or volume element.

For Riemannian manifolds

More specifically specifically, still, for(X,g)(X,g) an a (pseudo)-oriented (pseudo)-Riemannian manifold of dimension nn , itsvolume formthe vol g∈Ω n(X)vol_g \in \Omega^n(X)volume pseudoform is or thedifferential formvolume element of degreenvol g n vol_g which is measures a the specific volume differential as seen by the metric g n g n - in that it is characterized by any of the following equivalent statements.pseudoform that measures the volume as seen by the metric gg. If XX is oriented, then we may interpret vol g∈Ω n(X)vol_g \in \Omega^n(X) as a differential nn-form, also denoted vol gvol_g.

• The symmetric square vol g⋅vol gvol_g \cdot vol_g is equal to the nn-fold wedge product g∧⋯∧gg \wedge \cdots \wedge g, as elements of Ω n(X)⊗Ω n(X)\Omega^n(X) \otimes \Omega^n(X), and vol gvol_g is positive.

• The volume form is the image under the Hodge star operator ⋆ g:Ω k(X)→Ω n−k(X)\star_g\colon \Omega^k(X) \to \Omega^{n-k}(X) of the smooth function 1∈Ω 0(X)1 \in \Omega^0(X)

  vol g=⋆ g1. vol_g = \star_g 1 \,.

• In local oriented coordinates, vol g=|det(g)|vol_g = \sqrt{|det(g)|}, where det(g)det(g) is the determinant of the matrix of the coordinates of gg.

• For (E,Ω):TX→iso(n)(E, \Omega)\colon T X \to iso(n) the Lie algebra valued differential form on XX with values in the Poincare Lie algebra iso(n)iso(n) that encodes the metric and orientation (the spin connection Ω\Omega with the vielbein EE), the volume form is the image of (E,Ω)(E,\Omega) under the canonical volume Lie algebra cocycle vol∈CE(iso(n))vol \in CE(iso(n)):

  vol g=vol(E). vol_g = vol(E) \,.

  See Poincare Lie algebra for more on this.

This vol gvol_g is characterized by any of the following equivalent statements:

For unoriented manifolds

• The symmetric square vol g⋅vol gvol_g \cdot vol_g is equal to the nn-fold wedge product g∧⋯∧gg \wedge \cdots \wedge g, as elements of Ω n(X)⊗Ω n(X)\Omega^n(X) \otimes \Omega^n(X), and vol gvol_g is positive (meaning that its integral on any open submanifold? is nonnegative).

• The volume form is the image under the Hodge star operator ⋆ g:Ω k(X)→Ω n−k(X)\star_g\colon \Omega^k(X) \to \Omega^{n-k}(X) of the smooth function 1∈Ω 0(X)1 \in \Omega^0(X)

  vol g=⋆ g1. vol_g = \star_g 1 \,.

• In local oriented coordinates, vol g=|det(g)|vol_g = \sqrt{|det(g)|}, where det(g)det(g) is the determinant of the matrix of the coordinates of gg. In the case of a Riemannian (not pseudo-Riemannian) metric, this simplifies to vol g=det(g)vol_g = \sqrt{\det(g)}. (Note that local coordinates for a pseudoform include a local orientation, so this makes sense regardless of whether XX is oriented.)

• For (E,Ω):TX→iso(n)(E, \Omega)\colon T X \to iso(n) the Lie algebra valued differential form on XX with values in the Poincare Lie algebra iso(n)iso(n) that encodes the metric and orientation (the spin connection Ω\Omega with the vielbein EE), the volume form is the image of (E,Ω)(E,\Omega) under the canonical volume Lie algebra cocycle vol∈CE(iso(n))vol \in CE(iso(n)):

  vol g=vol(E). vol_g = vol(E) \,.

  See Poincare Lie algebra for more on this.

If (X,g)(X,g) is unoriented, then the definitions above don’t quite make sense, but (at least) the first three can be reinterpreted to give an nn-pseudoform instead, the volume pseudoform, or volume element. (Whether a pseudoform is positive is independent of the orientation, the Hodge operator still takes forms to pseudoforms, and local coordinates for a pseudoform include a local orientation.)

Degenerate cases

The If concept we of allow a volume (pseudo)-form (pseudo)form can to also be axiomatised: degenerate, on then any most of this goes through unchanged. In particular, a degenerate (pseudo)-nnRiemannian metric -dimensional manifold, defines a degenerate volume pseudoform (and hence a degenerate volume form on an oriented manifold).volume form is any nn-form that is positive (meaning that its integral on any open submanifold is nonnegative) and nondegenerate (meaning that it’s nowhere zero). Altogether, we may say that its integral on any inhabited open submanifold is (strictly) positive. Such a structure is actually equivalent to an absolutely continuous positive Radon measure on XX.

If However, a degenerate X n X n -form is unoriented, then we can only integrate n ω n \omega -pseudoforms, does not specify an orientation in general, so we there can is only not axiomatise necessarily the a concept good notion of volume pseudoform. form on an unoriented manifold. On the other hand, if we the equip open submanifold on which X ω≠0 X \omega \ne 0 with is a nondegeneratenndense -form , then there is at most one compatible orientation, although there still may be none. Of course, on an oriented manifold, forms are equivalent to pseudoforms, so we still know what a degenerate volume form is there.volvol, then there is a unique orientation oo on XX such that volvol is positive and therefore (relative to oo) a volume form. (Nondegeneracy is vital here.)

If To I remove remember the correctly, requirement every of volume positivity (pseudo)form comes from a metric, which is unique much iff more drastic; an arbitraryn≤1 n \leq 1 . -pseudoform is simply a11-density (and an arbitrary nn-form is a 11-pseudodensity). This is at best a notion of signed volume, rather than volume.

Properties

Since an nn-(pseudo)form is positive iff its integral on any open submanifold? is nonnegative and nondegenerate iff its integral on sufficiently small inhabited? open submanifolds is nonzero, a volume (pseudo)form may be defined as one whose integral on any inhabited open submanifold is (strictly) positive.

A volume (pseudo)form is also equivalent to an absolutely continuous positive Radon measure on XX. Here, nondegeneracy corresponds precisely to absolute continuity.

If I remember correctly, every volume (pseudo)form comes from a metric, which is unique iff n≤1n \leq 1.

• density

• divergence