In $\Bbb R^4$ we have curves, bi-dimensional surfaces and hypersurfaces. We integrate vector fields along curves and in hypersurfaces using a normal direction. I know that the adequate object to integrate in bi-dimensional surfaces are skew-symmetric matrix fields, but I'm unsure of how to do that. I saw these notes but I can't make the relation with differential forms.
I'll use the isomorphisms given in pages $264$ and $265$ of Munkres' Analysis on Manifolds (I'll add them in the end to avoid clutter).
Let $M^2 \subseteq \Bbb R^4$. The volume form of $M$ is expressed as $${\rm d}M = \sum_{1 \leq i<j\leq 4} (-1)^{i+j-1}\begin{vmatrix} n_i & n_j \\ \nu_i & \nu_j \end{vmatrix} \,{\rm d}x^1 \wedge \cdots \wedge \widehat{{\rm d}x^i} \wedge \cdots \wedge \widehat{{\rm d}x^j}\wedge \cdots \wedge {\rm d}x^4,$$where $\{n,\nu\}$ is a basis for the normal space, and it satisfies $$\begin{vmatrix} n_i & n_j \\ \nu_i & \nu_j \end{vmatrix}\,{\rm d}M = (-1)^{i+j-1}{\rm d}x^1 \wedge \cdots \wedge \widehat{{\rm d}x^i} \wedge \cdots \wedge \widehat{{\rm d}x^j}\wedge \cdots \wedge {\rm d}x^4$$along $M$, for $i<j$.
Question: Given a skew-symmetric matrix field $H$ over $M$, is there a neat function $f_{H,n,\nu}$ such that $f_{H,n,\nu}\,{\rm d}M = \gamma_2(H)$?
The obvious guess $\langle n,H\nu\rangle$ doesn't seems to work, we have some $(-1)^{i+j-1}$ remaining. To make it clear, I want some relation like $\langle F,T\rangle\,{\rm d}s = \alpha_1(F)$ for curves or $\langle F,n\rangle \,{\rm d}S = \beta_{n-1}(F)$ in hypersurfaces. This would allow us to get two Stokes-like formulas using the operator twist and spin mentioned in the notes I linked above.
I guess there must be some confusion on "integration over surfaces" versus "integration over submanifolds of codimension $2$", since $M^2 \subseteq \Bbb R^4$ is the problematic case. This raises more
Questions: is there more than one good way to identify skew-symmetric matrix fields with $2$-forms? Is there more than one way to define integration along surfaces in $\Bbb R^4$? Can we define yet different "twist" and "spin" operators from the ones given in the link?
I tried to not leave it too broad. Thanks.
For convenience:
- $\alpha_1:{\frak X} \to \Omega^1$, $\alpha_1(F) = \sum F_i\,{\rm d}x^i$;
- $\gamma_2: \text{skew-symmetric matrix fields}\to \Omega^2$, $\gamma_2(H) = \sum_{i<j}h_{ij}\,{\rm d}x^i\wedge {\rm d}x^j$;
- $\beta_{n-1}:{\frak X}\to \Omega^{n-1}$, $\beta_{n-1}(F) = \sum (-1)^{i-1}F_i\,{\rm d}x^1 \wedge \cdots \wedge \widehat{{\rm d}x^i}\wedge \cdots \wedge {\rm d}x^n$.