Motivation
In "Gelfand Transforms and Crofton formulas" the authors state that "Roughly speaking, densities are the most general objects that can be integrated over submanifolds independently of parameterization and orientation". So I wanted to understand these concepts better. There was a very good answer about densities (of rank $n$) here, and I was wondering how this extends to densities of rank $k$.
Density of rank $k$
In "Gelfand Transforms and Crofton formulas" the authors define a $k$-density (a density of rank $k$, not to be confused with the weight) over a vector space $V$ as a continuous function $\phi:\underbrace{V\times\ldots\times V}_k\to\mathbb{R}$ such that for any set of $k$ linearly independent vectors $\{v_j\}_{j=1}^k$ and $A\in GL(k,\mathbb{R})$ we have $$\phi(Av) = |\det A|\phi(v).$$ If I understood it correctly, the notation $Av$ means $(\sum_j A_{j1} v_j, \ldots, \sum_j A_{jk} v_j)$.
It is clear that for $k=n=\dim V$ this is just a usual (volume) density. For $k<n$, $\phi$ behaves like a density when restricted to any $k$-dimensional subspace of $V$. In fact it is less constrained than an $n$-density on the full space. Did I understand this correctly?
Examples of $k$-densities: $|\alpha|$, $\langle \alpha, \beta\rangle$?
Let $\alpha = \alpha_{i_1\ldots i_n} \, e^1\wedge\ldots \wedge e^n$ be a $n$-covector for an $n$-dimensional space. Then from $\alpha(Av) = (\det A) \,\alpha(v)$, and thus $|\alpha(Av)| = |\det A| \, |\alpha(v)|$ is a density. That is, it is enough to take the absolute value of an $n$-covector to get an $n$-density. If $V$ is a space over the complex numbers then $|z| = z^*z$. It's also clear that I can get a density of weight $s$ from $|\alpha|^s$.
Now let $\beta = \beta_{i_1\ldots i_n} \,e^1\wedge\ldots \wedge e^n$ then I believe the following object is an $n$-density of weight $2$: $$\langle\alpha,\beta\rangle = \alpha^*_{i_1\ldots i_n}G_{i_1\ldots i_n}\beta_{i_1\ldots i_n} (e^1\wedge\ldots\wedge e^n)^*(e^1\wedge\ldots\wedge e^n) = \alpha^*_{i_1\ldots i_n}G_{i_1\ldots i_n}\beta_{i_1\ldots i_n}|e^1\wedge\ldots\wedge e^n|^2.$$ This should hold because $$|(e^1\wedge\ldots \wedge e^n)(Av)|^2 = |\det A|^2\, |(e^1\wedge\ldots \wedge e^n)(v)|^2.$$
So if I want to consider $k<n$ I can try to see whether the above approaches work for $k$-densities.
Proof that $|\alpha|$ is a $k$-density
Let $\alpha = \sum_{|I|=k} \alpha_I \, (e^{i_1}\wedge\ldots\wedge e^{i_k})$. I want to show that $|\alpha(Av)| = |\det A| |\alpha(v)|$. I will show $\alpha(Av) = (\det A) \alpha(v)$ from which the former will follow.
Let $\{e_j\}_{j=1}^n$ be the basis of $V$ that is biorthogonal to the basis $\{e^i\}_{i=1}^n$ for $V^*$, biorthogonal meaning $e^i(e_j) = \delta^i_j$. Now let $\{v_r\}_{r=1}^k$ be $k$ vectors from $V$, then their coordinates with respect to $\{e_i\}_{i=1}^n$ are given as the $n\times k$ matrix $C_{ij} = e^i(v_j)$. Then it follows that: \begin{equation} \begin{aligned} \alpha(Av) &= \sum_{|I|=k}\alpha_I \,(e^{i_1}\wedge\ldots\wedge e^{i_k})(Av) \\ &=\sum_{|I|=k}\alpha_I \, \det C_IA \\ &= \sum_{|I|=k}\alpha_I \, \det C_I \det A \\ &= (\det A)\, \alpha(v). \end{aligned} \end{equation}
Here $C_I$ was the $k\times k$ submatrix of $C$ with rows from $I$ (in ascending order). Taking powers $|\alpha|^s$ results in a $k$-density of weight $s$.
Proof that $\langle \alpha, \beta \rangle$ is a $k$-density
Let $\alpha = \sum_{|I|=k} \alpha_I\, (e^{i_1}\wedge\ldots\wedge e^{i_k})$ and $\beta = \sum_{|J|=k} \beta_J\, (e^{j_1}\wedge\ldots\wedge e^{j_k})$. I define their "inner product" by distributing the terms: \begin{equation} \begin{aligned} \langle \alpha, \beta\rangle &= \left(\sum_{|I|=k}\alpha_I\,(e^{i_1}\wedge\ldots\wedge e^{i_k})\right)^*G \left(\sum_{|J|=k}\beta_J\,(e^{j_1}\wedge\ldots\wedge e^{j_k})\right)\\ &= \sum_{|I|=k}\sum_{|J|=k} \alpha_I^*G_{I,J}\beta_J \, (e^{i_1}\wedge\ldots\wedge e^{i_k})^*(e^{j_1}\wedge\ldots\wedge e^{j_k}), \end{aligned} \end{equation} where $^*$ is complex conjugation. The product is to be understood pointwise and not as a wedge or something else, i.e. $$\bigl((e^{i_1}\wedge\ldots\wedge e^{i_k})^*(e^{j_1}\wedge\ldots\wedge e^{j_k})\bigr)(v) = \bigl((e^{i_1}\wedge\ldots\wedge e^{i_k})(v)\bigr)^*\cdot\bigl((e^{j_1}\wedge\ldots\wedge e^{j_k})(v)\bigr),$$ where $\cdot$ is multiplication of complex numbers.
Now let $C_{ij} = e^i(v_j)$, then I can write: \begin{equation} \begin{aligned} \langle \alpha, \beta \rangle(Av) &= \left(\sum_{|I|=k}\alpha_I \,(e^{i_1}\wedge\ldots\wedge e^{i_k})(Av)\right)^*G\left(\sum_{|J|=k}\beta_J \,(e^{j_1}\wedge\ldots\wedge e^{j_k})(Av)\right) \\ &= \left(\sum_{|I|=k}\det \alpha_I \,\det C_IA\right)^*G\left(\sum_{|J|=k}\det \beta_J \,\det C_JA\right) \\ &= \left(\sum_{|I|=k}\alpha_I \,\det C_I \det A\right)^*G\left(\sum_{|J|=k}\det \beta_J \,\det C_J \det A\right) \\ &= (\det A)^*(\det A) \left(\sum_{|I|=k}\alpha_I \,(e^{i_1}\wedge\ldots\wedge e^{i_k})(v)\right)^*G\left(\sum_{|J|=k}\beta_J \,(e^{j_1}\wedge\ldots\wedge e^{j_k})(v)\right) \\ &= |\det A|^2 \langle \alpha, \beta\rangle (v). \end{aligned} \end{equation}
I can take powers $\langle \alpha,\beta\rangle^{s/2}$ to get densities with weight $s$. I should also be able to formulate a similar operation such as $\langle \alpha_1,\ldots,\alpha_r\rangle$ by expanding things in a similar manner, which should yield a $k$-density of weight $r$ (provided $r$ is even).
Interpretation
If $|\omega|$ is indeed a density for $k<n$, then what would be the physical/geometrical meaning of it in an integration problem? For instance a $2$-form in 3D $\omega = \star (f_1dx + f_2 dy + f_3 dz)$ can be used to integrate surface flux given a vector field $f$. It results in $f\cdot \hat{n} dS$. The corresponding $2$-density $|\omega|$ then integrates $|f\cdot \hat{n}|dS$, which is some kind of quantity that "doesn't care" about whether $f$ is in the positive or negative halfspace w.r.t. $n$ - it always takes the normal that would produce a positive result. Are there physical examples of something that would integrate like this over a surface? Some kind of non-oriented flux?
Densities of the form $\|\omega\|_2 = \sqrt{\langle \omega, \omega\rangle}$ should represent $k$-volume measure I believe. As an example for $k=1$ and $G_{ij} = \delta_{ij}$ I have $vol_1 = \sqrt{\sum_i (e^i)^2}$. So $G_{ij}$ here plays the role of metric tensor coefficients I guess, and the variants with more slots are some kinds of generalizations of this. I have no idea what $\langle \alpha, \beta\rangle$ for $\alpha\ne \beta$ is though.
Double Cover and Densities
Finally, if I were to integrate a $2$-form $\omega$ over the double cover of a non-orientable surface, e.g. a Mobius strip in $\mathbb{R}^3$, is it equivalent to twice the integral of $|\omega|$ or $\|\omega\|_2$ or some other $2$-density? Or is it something else? Is there a theorem about this anywhere?
More generally, is the correspondence: form $\to$ oriented manifold, pseudoform $\to$ potentially non-oriented but orientable manifold, density $\to$ potentially non-orientable manifold.
