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?

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    $\begingroup$ I heard this has been reformulated in the language of "densities". So you can integrate a density on a mobius band, for example. $\endgroup$ Aug 23 '19 at 19:33
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    $\begingroup$ IIRC a "density" is not quite the same as a "twisted differential form" or "pseudoform" (ncatlab.org/nlab/show/differential+form#twisted), which might be another name for "odd type" forms. $\endgroup$ Aug 23 '19 at 22:11
  • $\begingroup$ @MikeShulman twisted forms was the first thing that came to my mind, since I work very much with them. Still I could not make the precise link between them and de Rham's odd type forms. $\endgroup$
    – Alan Muniz
    Aug 24 '19 at 1:07
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    $\begingroup$ Well, in that case maybe you should include the definition of "odd type form" in the question. $\endgroup$ Aug 24 '19 at 2:25

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.


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