It's well known that one can integrate a differential form on an oriented submanifold. Less well known (but also true), one can integrate a differential pseudoform on an pseudoriented (transversely oriented) submanifold. But in classical differential geometry, one also sees forms that can be integrated on unoriented submanifolds.
I call these absolute forms. The term ‘absolute’ suggests a lack of additional required structure, in this case some sort of orientation on the domain of integration. It also suggests absolute value, since many of the examples from classical differential geometry involve absolute values. Indeed, we can define the absolute value of a form or a pseudoform to be an absolute form, although not every absolute form arises in this way.
The main theorem of absolute forms is that, if is a (pseudo)--form and is a (pseudo)-oriented -dimensional submanifold, then
where is an absolute -form (the absolute value of ), is simply with its (pseudo)-orientation ignored, and the absolute value on the left is the ordinary absolute value of scalars. This theorem also applies if we start with an absolute -form , (although in that case starts out unoriented and so is the same as ). If is a de Rham chain (a formal linear combination of appropriately oriented submanifolds), we also take absolute values of the formal coefficients in .
Let be a differentiable manifold (or similar sort of space), and let be a natural number (typically , where is the dimension of ). Recall that an (exterior differential) -form on is a function that assigns a real number (or whatever is the relevant sort of scalar) to a point in and a -tuple of tangent vectors at , multilinearly? and alternating? in the . Similarly, a -pseudoform on is a function that assigns a scalar to a point in , a local orientation at , and a -tuple of tangent vectors at , multilinearly and alternating in the and reversing sign under a reversal of .
Fixing , shall be uniformly continuous.
Fix a -dimensional subspace of the tangent space at and an orientation of . Now given a linearly independent -tuple from (that is a basis of ), let be according to whether the orientation of induced by the matches , and extend this by continuity to all -tuples from (which extension must be unique and exists by 1&2). The resulting function shall be multilinear? (and so also alternating, by 2).
The multilinearity condition here is rather weaker than for a (pseudo)-form, since it applies only within a -dimensional subspace. Shifting one vector even slightly outside of loses all connection provided by multilinearity, which is why we need a continuity condition; continuity holds for (pseudo)-forms automatically.
An absolute -form is continuous if it is jointly continuous in all of its data ( as well as the ). Since the domain of the function is a manifold (a vector bundle over , although is not a map of vector bundles), we can even discuss differentiability, smoothness, and even analyticity of when has the relevant structure.
An absolute -form is the same thing as a -form. An absolute -form on an -dimensional manifold is essentially the same thing as an -pseudoform; with the notation from condition 3, the only possibility for is the entire tangent space , and we have
to relate the -pseudoform to the absolute -form . Finally, the only absolute -form for is .
At a point , an absolute -form is:
indefinite if for some (necessarily linearly independent) -tuple of vectors and for some -tuple,
semidefinite if not indefinite,
definite (and hence semidefinite) if for every independent -tuple of vectors at ,
positive (and hence semidefinite) if for every -tuple of vectors (it is enough when they are independent),
negative (and hence semidefinite) if for every (independent) -tuple of vectors.
All these are at a point ; satisfies the condition tout court if it holds for all .
Given an absolute -form , its absolute value is a positive semidefinite absolute -form:
If we start with a -form , then the same definition defines a positive absolute -form . If we start with a -pseudoform , then essentially the same definition still works; we use either orientation to evaluate with the same result. Note that is continuous if is. However, we may not conclude that is differentiable just because is differentiable (or even analytic). On the other hand, inherits differentiability properties from wherever . (Even then, however, we cannot inherit analyticity, except in dimension.)
Given two absolute -forms and , their sum is an absolute -form:
Given an absolute -form and a scalar field , their product is an absolute -form:
In this way, the space of absolute -forms is a module over the algebra of scalar fields and the space of sections of a vector bundle. For now, we decline to define products of absolute forms of aribtrary rank.
Given an absolute -form on , a manifold , and a continuously differentiable map , the pullback is an absolute -form on :
Here, is the pushforward? of under . Note that is continuous if is; we can also pull back differentiability and analyticity properties that and both have.
Given a continuous absolute -form on , a -dimensional manifold , and a continuously differentiable map , the integral is a scalar:
On the right-hand side, is a continuous absolute -form on , but since is -dimensional, this is essentially the same as a continuous -pseudoform on , and we already know how to integrate this (see integration of differential forms).
Examples of absolute forms from classical differential geometry include:
Absolute -forms are the same as ordinary -forms.
More generally, the arclength? element on a Riemannian manifold is an absolute -form. Neither nor (in general) is actually the differential of anything, but is the canonical vector-valued -form (which, on an affine space, really is the differential of the identity map ), and we really can use the metric to take the norm of such a form to get an absolute -form.
Similarly, the surface area? element on a Riemannian manifold is an absolute -form, and we can continue into higher dimensions (although the classical volume element in is already covered as a -pseudoform). In principle, we should have , etc; but it's not clear what this is in general.
Near the end of a Usenet post from 2002, we see a definition of for a (pseudo)--form and a -dimensional submanifold, but without a broader context for itself:
Apparently absolute -forms (at least if continuous) are the same as even -densities as defined by Gelfand; see this MathOverflow answer: