# nLab kernel of integration is the exact differential forms

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## 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

## 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 $d \colon \Omega^{n-1}_{cp} \to \Omega^n_{cp}(X)$.

###### 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.