I am trying to understand, in as simple terms as possible:
- How to define integration for non-orientable manifolds, and
- why it is impossible to do so using only differential forms.
In particular, I've seen some discussion of using "densities" instead of $n$-forms for integration, but am not really clear on why densities are required. In other words, is it really impossible to define integration on nonorientable manifolds using forms alone?
I am of course aware that any $n$-form must vanish somewhere on a nonorientable manifold, so we cannot find a volume form, hence cannot use the standard definition of integration. I think the reason I'm not finding this answer satisfying is that it is a bit tautological: we can't define integration with respect to volume forms because there are no volume forms. But why must we define integration with respect to a (global) volume form in the first place? Is there really no other way to do it using locally-defined forms? Thinking of a manifold as a collection of local charts is common in geometry, and I'm having trouble understanding why this approach doesn't work in the case of integration.
On an orientable manifold, we define integration of functions with respect to a volume form. On a non orientable manifold, there are no volume forms, so we have to do something else!