nLab kernel of integration is the exact differential forms

Idea

The Stokes theorem immediately implies that the integration of an exact differential form with compact support vanishes. But in fact also the converse is true: If the integration of a differential form with compact support of top degree vanishes, then it is exact (prop. below).

This statement underlies for instance

• higher Lie integration by the “path method”

• the interpretation of the BV-BRST complex as computing expectation values (see there).

Statement

Proposition

Let $X$ be an oriented connected smooth manifold of finite dimension. Let $n=\dim X$ and write $\Omega^n_{cp}(X)$ for the vector space of differential n-forms with compact support and

$$\int_X \;\colon\; \Omega^n_{cp}(X) \longrightarrow \mathbb{R}$$

for the linear map to the real numbers given by integration of differential forms.

Then the kernel of this map is precisely the exact differential forms

$$ker\left(\int_X\right) = im(d) \,,$$

hence the image of the de Rham differential $d \colon \Omega^{n-1}_{cp} \to \Omega^n_{cp}(X)$.

(e.g. Lafointaine 15, section 7.3, theorem 7.5)

Proposition

At least when $X = \mathbb{R}^n$ is a Cartesian space, then the statement of prop. also holds in smoothly indexed sets of smooth differential forms.

(e.g. Lafointaine 15, section 7.3, lemma 7.3)

Related concepts

• de Rham theorem

• Lie integration (proof of this prop.)

References

• Jacques Lafontaine, section 7.3 of An introduction to differential manifolds, Springer 2015 (doi:10.1007/978-3-319-20735-3)