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 $\omega$ is a (pseudo)-$p$-form and $R$ is a (pseudo)-oriented $p$-dimensional submanifold, then
where ${|\omega|}$ is an absolute $p$-form (the absolute value of $\omega$), $|{R}|$ is simply $R$ 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 $p$-form $\omega$, (although in that case $R$ starts out unoriented and so is the same as ${|R|}$). If $R$ is a de Rham chain (a formal linear combination of appropriately oriented submanifolds), we also take absolute values of the formal coefficients in ${|R|}$. (This operation does not respect the usual notion of equality of chains, but the theorem is true all the same.)
Let $X$ be a differentiable manifold (or similar sort of space), and let $p$ be a natural number (typically $0 \leq p \leq n$, where $n$ is the dimension of $X$). Recall that an (exterior differential) $p$-form $\omega$ on $X$ is a function that assigns a real number (or whatever is the relevant sort of scalar) $\omega_c(v_1,\ldots,v_p)$ to a point $c$ in $X$ and a $p$-tuple $(v_1,\ldots,v_p)$ of tangent vectors at $c$, multilinearly and alternating in the $v_i$. Similarly, a $p$-pseudoform $\omega$ on $X$ is a function that assigns a scalar $\omega_c^o(v_1,\ldots,v_p)$ to a point $c$ in $X$, a local orientation $o$ at $c$, and a $p$-tuple $(v_1,\ldots,v_p)$ of tangent vectors at $c$, multilinearly and alternating in the $v_i$ and reversing sign under a reversal of $o$.
An absolute $p$-form $\omega$ on $X$ is a function that assigns a scalar $\omega_c(v_1,\ldots,v_p)$ to a point $c$ in $X$ and a $p$-tuple $(v_1,\ldots,v_p)$ of tangent vectors at $c$ and that satisfies the following conditions:
Fixing $c$, $\omega_c({-})$ shall be uniformly continuous.
The $p$-tuple $(v_1,\ldots,v_p)$ shall be linearly independent if $\omega_c(v_1,\ldots,v_p) \ne 0$. Thus, although $\omega_c$ is not linear, we may still call it alternating; however (as a consequence of 3), it is actually symmetric.
Fix a $p$-dimensional subspace $S$ of the tangent space at $c$ and an orientation $o$ of $S$. Now given a linearly independent $p$-tuple $(v_1,\ldots,v_p)$ from $S$ (that is a basis of $S$), let $\omega_c(v_1,\ldots,v_p)_S^o$ be $\pm\omega_c(v_1,\ldots,v_p)$ according to whether the orientation of $S$ induced by the $v_i$ matches $o$, and extend this by continuity to all $p$-tuples from $S$ (which extension must be unique and exists by 1&2). The resulting function $\omega_c({-})_S^o$ 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 $p$-dimensional subspace. Shifting one vector even slightly outside of $S$ loses all connection provided by multilinearity, which is why we need a continuity condition; continuity holds for (pseudo)-forms automatically.
An absolute $p$-form $\omega$ is continuous if it is jointly continuous in all of its data ($c$ as well as the $v_i$). Since the domain of the function $\omega$ is a manifold (a vector bundle over $X$, although $\omega$ is not a map of vector bundles), we can even discuss differentiability, smoothness, and even analyticity of $\omega$ when $X$ has the relevant structure.
An absolute $0$-form is the same thing as a $0$-form. An absolute $n$-form on an $n$-dimensional manifold $X$ is essentially the same thing as an $n$-pseudoform; with the notation from condition 3, the only possibility for $S$ is the entire tangent space $T_c{X}$, and we have
to relate the $n$-pseudoform $\tilde{\omega}$ to the absolute $n$-form $\omega$. Finally, the only absolute $p$-form for $p \gt n$ is $0$.
At a point $c$, an absolute $p$-form $\omega$ is:
indefinite if $\omega_c(v_1,\ldots,v_p) \gt 0$ for some (necessarily linearly independent) $p$-tuple of vectors and $\omega_c(v_1,\ldots,v_p) \lt 0$ for some $p$-tuple,
semidefinite if not indefinite,
definite (and hence semidefinite) if $\omega_c(v_1,\ldots,v_p) \ne 0$ for every independent $p$-tuple of vectors at $c$,
positive (and hence semidefinite) if $\omega_c(v_1,\ldots,v_p) \geq 0$ for every $p$-tuple of vectors (it is enough when they are independent),
negative (and hence semidefinite) if $\omega_c(v_1,\ldots,v_p) \leq 0$ for every (independent) $p$-tuple of vectors.
All these are at a point $c$; $\omega$ satisfies the condition tout court if it holds for all $c$.
Given an absolute $p$-form $\omega$, its absolute value ${|\omega|}$ is a positive semidefinite absolute $p$-form:
If we start with a $p$-form $\omega$, then the same definition defines a positive absolute $p$-form ${|\omega|}$. If we start with a $p$-pseudoform $\omega$, then essentially the same definition still works; we use either orientation to evaluate $\omega$ with the same result. Note that ${|\omega|}$ is continuous if $\omega$ is. However, we may not conclude that ${|\omega|}$ is differentiable just because $\omega$ is differentiable (or even analytic). On the other hand, ${|\omega|}$ inherits differentiability properties from $\omega$ wherever $\omega \ne 0$. (Even then, however, we cannot inherit analyticity, except in $1$ dimension.)
Given two absolute $p$-forms $\omega$ and $\eta$, their sum $\omega + \eta$ is an absolute $p$-form:
Given an absolute $p$-form $\omega$ and a scalar field $f$, their product $f \omega$ is an absolute $p$-form:
In this way, the space of absolute $p$-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 $p$-form $\omega$ on $X$, a manifold $U$, and a continuously differentiable map $R\colon U \to X$, the pullback $R^*\omega$ is an absolute $p$-form on $U$:
Here, $R_*v_i$ is the pushforward of $v_i$ under $R$. Note that $R^*\omega$ is continuous if $\omega$ is; we can also pull back differentiability and analyticity properties that $\omega$ and $R$ both have.
Given a continuous absolute $p$-form $\omega$ on $X$, a $p$-dimensional manifold $U$, and a continuously differentiable map $R\colon U \to X$, the integral $\int_R \omega$ is a scalar:
On the right-hand side, $R^*\omega$ is a continuous absolute $p$-form on $U$, but since $U$ is $p$-dimensional, this is essentially the same as a continuous $p$-pseudoform on $U$, and we already know how to integrate this (see integration of differential forms).
Examples of absolute forms from classical differential geometry include:
Absolute $0$-forms are the same as ordinary $0$-forms.
Absolute $n$-forms on an $n$-dimensional manifold are the same as $n$-pseudoforms (and hence the same as absolutely continuous Radon measures).
In complex analysis, ${|\mathrm{d}z|}$ is an absolute $1$-form sometimes used in contour integration. This literally is the absolute value of the differential of the identity map $z$.
More generally, the arclength element $\mathrm{d}s = {\|\mathrm{d}\mathbf{x}\|}$ on a Riemannian manifold is an absolute $1$-form. Neither $\mathrm{d}s$ nor (in general) $\mathrm{d}\mathbf{x}$ is actually the differential of anything, but $\mathrm{d}\mathbf{x}$ is the canonical vector-valued $1$-form (which, on an affine space, really is the differential of the identity map $\mathbf{x}$), and we really can use the metric to take the norm of such a form to get an absolute $1$-form.
Similarly, the surface area? element $\mathrm{d}S$ on a Riemannian manifold is an absolute $2$-form, and we can continue into higher dimensions (although the classical volume element $\mathrm{d}V$ in $\mathbb{R}^3$ is already covered as a $3$-pseudoform). In principle, we ought to be able to write down expression for $\mathrm{d}S$ etc in terms of $\mathrm{d}s$, although so far the only thing that I know how to do is $\mathrm{d}S = {\|{\mathrm{d}\mathbf{x} \hat\times \mathrm{d}\mathbf{x}}\|} / 2$, where $\hat\times$ indicates a wedge product of vector-valued forms whose vectors are multiplied by the cross product. (This can be generalized to any finite-dimensional area in any finite-dimensional Riemannian manifold; in particular, $\mathrm{d}V = {|{\mathrm{d}\mathbf{x} \hat\cdot \mathrm{d}\mathbf{x} \hat\times \mathrm{d}\mathbf{x}}|} / 6$.)
The reason that $\mathrm{d}s$ determines $\mathrm{d}S$ is that lengths determine areas, as through Heron's Formula. However, our usual ways of deriving one form from another (thinking of them, at a given point, as maps from lists of vectors to numbers) involve only rerranging (permuting, duplicating, and contracting) the inputs and applying operations to the outputs. But this cannot be sufficient; knowing only the lengths of two vectors $v$ and $w$ (which vectors are the inputs to $\mathrm{d}S$, and which lengths are the outputs of $\mathrm{d}s$ at those inputs) does not tell you the area of the parallelogram that they span (which is the output of $\mathrm{d}S$). You also need the length of $v - w$ (or of $v + w$). So while we can express $\mathrm{d}S_c$ ($\mathrm{d}S$ at a point $c$) as $(v, w \mapsto 2 H(\mathrm{d}s(v), \mathrm{d}s(w), \mathrm{d}s(v - w))$ (where $H$ is the Heron function, the real-valued function of three real variables that takes the lengths of the three sides of a triangle and returns its area), or equivalently as $(v, w \mapsto 2 H({\|v\|}, {\|w\|}, {\|{v - w}\|})$, we cannot express $\mathrm{d}S$ with a formula anything along the lines of ${\|{\mathrm{d}\mathbf{x} \hat\times \mathrm{d}\mathbf{x}}\|} / 2$ using $\mathrm{d}s$ directly instead of $\mathrm{d}\mathbf{x}$. (And so you may as well describe $\mathrm{d}S_c$ as $(v, w \mapsto {\|{v \times w}\|})$ instead.)
exterior differential forms and pseudoforms (the more well-known variations);
cojet differential forms (and cogerm differential forms more generally) generalize both exterior and absolute $1$-forms and pseudo-$1$-forms;
coflare differential form?s are a common generalization of exterior forms, absolute forms, and cojet forms (although not all cogerm forms).
Near the end of a Usenet post from 2002, we see a definition of $\int_R {|\omega|}$ for $\omega$ a (pseudo)-$p$-form and $R$ a $p$-dimensional submanifold, but without a broader context for ${|\omega|}$ itself:
Apparently absolute $p$-forms (at least if continuous) are the same as even $p$-densities as defined by Gelfand; see this MathOverflow answer:
Last revised on December 3, 2019 at 22:57:47. See the history of this page for a list of all contributions to it.