I understand some of the mechanics behind integration over manifolds, but I am lacking on some of the intuition.
In Tu's An Introduction to Manifolds, when introducing integration of differential forms (Section 23.4), he says:
Our approach to integration over a general manifold has several distinguishing features:
- The manifold must be oriented
- On a manifold of dimension $n$, one can integrate only $n$-forms, not functions
- The $n$-forms must have compact support.
My current understanding is that (1) is necessary to give a sign to the integral, just as we have in $\mathbb{R}^n$.
(3) is necessary to eliminate cases where the domain is infinite (for example, to rule out $\int_{-\infty}^\infty dx$).
However, I am less certain about why (2) is necessary. In Lee's An Introduction to Smooth Manifolds, he gives some intuition about how top forms provide a way of measuring "signed volume" at each point of a manifold. Thus I see why they're useful in integration, but I don't see why this rules out integrating differential forms that are not top forms (including $0$ forms, i.e. functions). What is the intuition behind this?
Best Answer
The central idea motivating integration is to provide a way of "summing up" all the values of some function over a given region. This must be done through some limiting process, e.g. a Riemann sum, and in order for this limit to make any sense, we must have a concept of how much to weight an infinitesimally small "cube" in the region. This is done by defining some notion of volume. For an $n$-dimensional manifold, $n$-forms are a natural way to define volume. Therefore a reasonable notion of integration on a manifold is inherently tied to specifying some $n$-form.
Maybe a more lowbrow reason: think about intuition from multivariable calculus. To integrate over an $n$-dimensional cube, you iteratively take $n$ single variable integrals corresponding to the $n$ coordinate directions, and for this to work you really do need to be integrating an $n$-form. I can't imagine a reasonable definition where you could integrate with fewer or more forms, since that would no longer capture the idea of "integrating over an $n$-dimensional cube."