Question

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.

Answer 1

There at least two sources I am aware of.

1. Theodore Frankel, The Geometry of Physics, Section 2.8 and 3.4.

2. Georges De Rham, Varietes Differentiables. Formes, courants, formes harmoniques, Chap. II.

Answer 2

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.