nLab
closed differential form

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

A differential form ωΩ p(X)\omega \in \Omega^p(X) is called closed if the de Rham differential d:Ω p(X)Ω p+1(X)d \colon \Omega^{p}(X) \to \Omega^{p+1}(X) sends it to zero: dω=0d \omega = 0, hence if it is in the kernel of the de Rham.

A differential form ωΩ p+1(X)\omega \in \Omega^{p+1}(X) is called exact if it is in the image of the de Rham differential: ω=dα\omega = d \alpha, for some αΩ p(X)\alpha \in \Omega^{p}(X).

The quotient of the vector space of closed differential forms by the exact differential forms of degree pp is the de Rham cohomology of XX in degree pp.

Formalization of closed and co-exact differential forms in cohesive homotopy theory is discussed at differential cohomology hexagon.

Properties

Last revised on October 30, 2017 at 15:55:11. See the history of this page for a list of all contributions to it.