If the manifold is unoriented, then we get a volume pseudoform instead, or equivalently a volume density (of weight ). We can also consider volume (pseudo)-forms in the absence of a metric, in which case we have a choice of volume forms.
For a general smooth manifold of finite dimension , a volume form on is a nondegenerate (nowhere vanishing) differential -form on , equivalently a nondegenerate section of the canonical line bundle? on . A volume pseudoform or volume element on is a positive definite density (of rank ) on , or equivalently a positive definite differential -pseudoform on .
A volume form defines an orientation on , the one relative to which it is positive definite. If is already oriented, then we require the orientations to agree (to have a volume form on 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.
More specifically, for a (pseudo)-Riemannian manifold of dimension , the volume pseudoform or volume element is a specific differential -pseudoform that measures the volume as seen by the metric . If is oriented, then we may interpret as a differential -form, also denoted .
This is characterized by any of the following equivalent statements:
vol_g = \star_g 1 \,.
In local oriented coordinates, , where is the determinant of the matrix of the coordinates of . In the case of a Riemannian (not pseudo-Riemannian) metric, this simplifies to . (Note that local coordinates for a pseudoform include a local orientation, so this makes sense regardless of whether is oriented.)
For the Lie algebra valued differential form on with values in the Poincare Lie algebra that encodes the metric and orientation (the spin connection with the vielbein ), the volume form is the image of under the canonical volume Lie algebra cocycle :
vol_g = vol(E) \,.
See Poincare Lie algebra for more on this.
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 -form 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 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 -pseudoform is simply a -density (and an arbitrary -form is a -pseudodensity). This is at best a notion of signed volume, rather than volume.
Since an -(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.
If I remember correctly, every volume (pseudo)form comes from a metric, which is unique iff .