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](https://zbmath.org/?q=an:0284.58001).


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?



Odd differential forms
A form of odd type is the same thing as a [pseudoform](https://ncatlab.org/nlab/show/differential+form#twisted), 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](https://arxiv.org/abs/math/0101268). 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.


