Odd differential forms

Question

In de Rham's classical book "Variétés Différentiables"

de Rham, Georges, Variétés différentiables. Formes, courants, formes harmoniques. 3e éd. revue et augmentée, Publications de l’Institut de Mathématique de l’Université de Nancago III. Actualités scientifiques et industrielles 1222 b. Paris: Hermann. X, 198 p. (1973). ZBL0284.58001.

it is defined the concept of an differential form of "odd type" that carries a sign and makes sense in non-oriented(able) manifolds.

Since we usually only encounter so called forms of "even type" in modern standard textbooks, I was wondering what further developments these odd forms had in the past decades.

Then my question is: are these forms of odd type still studied? Do they have any further applications?

Answer 1

A form of odd type is the same thing as a pseudoform, as mentioned by Mike Shulman. This is a form twisted by the pseudo-scalar bundle $\Psi$. The issue with the term "density" is that it has multiple meanings. It can refer to forms twisted by $\Psi$ or by $\bigwedge^{\!n}T^*\!X$ (where $X$ is an $n$-dimensional manifold). These bundles are isomorphic, but not canonically so, because the isomorphism depends on a choice of Riemannian metric. I think that pseudo-forms are more natural objects to integrate and to multiply. As described in de Rham, we get a system of multiplication for even and odd forms, where the parity of multiplication looks like addition in $\mathbb Z/2\mathbb Z$. This is because $\Psi\otimes \Psi$ has a canonical trivialization, so the product of two $\Psi$-twisted forms can be identified with a form twisted by the trivial line bundle, i.e. our usual notion of a differential form.

Unfortunately, I don't have the knowledge/expertise to point to various present applications, so this answer is somewhat incomplete. But I can say how I stumbled upon it: I am reading de Rham in order to understand currents, so that I can read Harvey and Lawson - Finite volume flows and Morse theory. Currents are a generalization of forms of even and odd type, allowing us to integrate "Schwartz distributions" on manifolds. The paper I mentioned shows that pulling back a differential form by a nice enough flow will limit towards a current as $t\rightarrow \infty$. The authors use this notion to provide a beautiful description of Morse homology.

Answer 2

For a connect smooth manifold $M$, there is an orientation covering $\pi:\tilde M\to M$, where $M$ is a two-copy of $M$ if $M$ is orientable and is a connected orientable manifold if $M$ is nonorientable. Anyway, the Deck transformation group of $\pi$ is $\{\mathrm{Id},\tau\}\simeq\mathbb Z/2\mathbb Z$, which acts also on the space of differential forms $\Omega^\bullet(\tilde M)$ by pullback. Then $\Omega^\bullet(\tilde M)$ can be decomposed into direct sum of two subspaces: $\Omega_{\mathrm {even}}^\bullet(\tilde M)$ and $\Omega_{\mathrm{odd}}^\bullet(\tilde M)$, where $$\Omega_{\mathrm {even}}^\bullet(\tilde M) = \{\omega\in\Omega^\bullet(\tilde M)|\tau^*\omega = \omega\},$$ $$\Omega_{\mathrm {odd}}^\bullet(\tilde M) = \{\omega\in\Omega^\bullet(\tilde M)|\tau^*\omega = -\omega\}.$$ We can identify ''even forms'' and ''odd forms'' in the sense of de Rham with elements in $\Omega_{\mathrm {even}}^\bullet(\tilde M)$ and $\Omega_{\mathrm {odd}}^\bullet(\tilde M)$, respectively.