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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. $\endgroup$ – Bombyx mori Aug 23 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. $\endgroup$ – Mike Shulman Aug 23 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 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. $\endgroup$ – Mike Shulman Aug 24 at 2:25

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