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kernel of integration is the exact differential forms

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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}{}& ʃ &\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 opposite is true: If the integration of a differential form with compact support of top degree vanishes, then it is exact (prop. below) .

Statement

Proposition

Let XX be a smooth manifold of dimension n1n \geq 1. Write Ω cp n(X)\Omega^n_{cp}(X) for the vector space of differential n-forms with compact support and

X:Ω cp n(X) \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( X)=im(d), ker\left(\int_X\right) = im(d) \,,

hence the image of the de Rham differential d:Ω cp n1Ω cp n(X)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= nX = \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)

References

  • Jacques Lafontaine, section 7.3 of An introduction to differential manifolds, Springer 2015

Created on October 30, 2017 at 15:48:27. See the history of this page for a list of all contributions to it.