basic differential form


Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

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



smooth space


The magic algebraic facts




tangent cohesion

differential cohesion

graded differential cohesion

singular 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}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



Via pullback to smooth principal bundles

Given a Lie group GG and a smooth GG-principal bundle PpXP \overset{p}{\longrightarrow} X over a base smooth manifold XX, a differential form ωΩ (P)\omega \in \Omega^\bullet\big( P \big) on the total space PP is called basic if it is the pullback of differential forms along the bundle projection pp of a differential form βΩ (X)\beta \in \Omega^\bullet\big( X \big) of the base manifold

ω=p *(β). \omega \;=\; p^\ast(\beta) \,.

In terms of Cartan calculus

Equivalently, if 𝔤T eG\mathfrak{g} \simeq T_e G denotes the Lie algebra of GG, and for v𝔤v \in \mathfrak{g} we write

v^:P(e,v),(,0)TG×TPT(G×P)dρTP \hat v \;\colon\; P \overset{ (e,v), (-,0) }{\hookrightarrow} T G \times T P \simeq T ( G \times P ) \overset{ \;\;\; d \rho \;\;\; }{\longrightarrow} T P

for the vector field on PP which is the derivative of the GG-action G×PρPG \times P \overset{\rho}{\to} P along vv, then differential form ω\omega is basic precisely if

  1. it is annihilated by the contraction with v^\hat v

    ι v^ω=0 \iota_{\hat v} \omega = 0
  2. it is annihilated by the Lie derivative along v^\hat v:

    v^ω=[d dR,ι v^]ω=0 \mathcal{L}_{\hat v}\omega \;=\; [d_{dR}, \iota_{\hat v}] \omega \;=\; 0

    (where the first equality holds generally by Cartan's magic formula, we are displaying it just for emphasis)

for all v𝔤v \in \mathfrak{g}.

In this form the definition of basic forms makes sense more generally whenever a Cartan calculus is given, not necessarily exhibited by smooth vector fields on actual manifolds. This more general concept of basic differential forms appears notably in the construction of the Weil mdoel for equivariant de Rham cohomology.