nLab invariant 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




infinitesimal cohesion

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)



Invariant differential form

A differential form ωΩ dR p(G)\omega \in \Omega_{dR}^p(G) on a Lie group GG is called left invariant if for every gGg \in G it is invariant under the pullback of differential forms

(1)(L g) *ω=ω (L_g)^* \omega = \omega

along the left multiplication action

L g: G G x gx \array{ L_g \colon & G &\longrightarrow& G \\ & x &\mapsto& g \cdot x }

Analogously a form is right invariant if it is invariant under the pullback by right translations R gR_g.

More generally, given a differentiable (e.g. smooth) group action of GG on a differentiable (e.g. smooth) manifold MM

G×M ρ M (g,x) gx \array{ G \times M & \overset{\rho}{\longrightarrow} & M \\ (g,x) &\mapsto& g \cdot x }

then a differential form ωΩ dR p(M)\omega \in \Omega^p_{dR}(M) is called invariant if for all gGg \in G

ρ(g) *(ω)=ω. \rho(g)^\ast(\omega) \;=\; \omega \,.

This reduces to the left invariance (1) for M=GM = G and ρ\rho being the left multiplication action of GG on itself.

Invariant vector field

For a vector field XX one instead typically defines the invariance via the pushforward (TL g)X=(L g) *X(T L_g) X = (L_g)_* X. Regarding that L gL_g and T gT_g are diffeomorphisms, both pullbacks and pushfowards (hence invariance as well) are defined for every tensor field; and the two requirements are equivalent.


Last revised on April 27, 2024 at 10:22:29. See the history of this page for a list of all contributions to it.