I'm looking for a definition of pseudo differential forms in ordinary differential 
geometry. However searching the web gave me nothing. There are definitions in supergeometry
but that is not what I'm after.


Recently I read, that pseudo-differentialforms are the natural structure to integrate, 
since integration works on any kind of submanifold (orientation not required) for them, 
but those texts don't gave a 'clean' definition of these kind of forms.


What are pseudo-differentialforms?


Can pseudo differentialforms be defined as sections of some kind of fiber bundle?
If yes that's a definition I would prefer. 



Pseudo-Differentialforms
Pseudo-Forms:


Let $M$ be a topological manifold and $PM$ the frame bundle of $M$. If $dim(M)=n$ then $PM$
is a $Gl(n)$-principal bundle.


Let $\tau: Gl(n) \to \mathbb{R} \; ; \; A \mapsto abs(det(A))$ the map, that maps any linear
isomorphism $f \in Gl(n)$ to the absolute value of its determinant. This defines a left 
action of $Gl(n)$ on $\mathbb{R}$ by


$$\cdot: Gl(n) \times \mathbb{R} \to \mathbb{R} \; ; \; (A,x) \mapsto \tau(A)x$$


The bundle of pseudo-forms then is the associated (line) bundle 


$$PM[\mathbb{R},\cdot]$$ 


of this action and pseudo-forms are sections of this bundle. If $M$ is smooth, this is a smooth bundle,since
the action is smooth. ($\tau$ is smooth since $det(A)\neq0$ for $A\in Gl(n)$)


But this gives only pseudo-forms that behaves right in respect to integration on $dim(M)$-dimensional submanifolds. Remains the question, ow to generalize this to submanifolds of arbitrary dimension. 


