nLab
volume form

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Riemannian geometry

Differential geometry

differential geometry

synthetic differential geometry

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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, for (X,g)(X,g) a (pseudo)-Riemannian manifold of dimension nn, the volume pseudoform or volume element vol gvol_g is a specific differential nn-pseudoform that measures the volume as seen by the metric gg. If XX is oriented, then we may interpret vol gΩ n(X)vol_g \in \Omega^n(X) as a differential nn-form, also denoted vol gvol_g.

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

  • The symmetric square vol gvol gvol_g \cdot vol_g is equal to the nn-fold wedge product ggg \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 nk(X)\star_g\colon \Omega^k(X) \to \Omega^{n-k}(X) of the smooth function 1Ω 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):TXiso(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 volCE(iso(n))vol \in CE(iso(n)):

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

    See Poincare Lie algebra for more on this.

Degenerate cases

If we allow a volume (pseudo)form to be degenerate, then most of this goes through unchanged. In particular, a degenerate (pseudo)-Riemannian metric defines a degenerate volume pseudoform (and hence a degenerate volume form on an oriented manifold).

However, a degenerate nn-form ω\omega does not specify an orientation in general, so there is not necessarily a good notion of volume form on an unoriented manifold. On the other hand, if the open submanifold on which ω0\omega \ne 0 is dense, 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.

To remove the requirement of positivity is much more drastic; an arbitrary nn-pseudoform is simply a 11-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 n1n \leq 1.

Revised on January 25, 2013 23:01:55 by Toby Bartels (64.89.53.16)