nLab
invariant 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

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}{}& 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

Definition

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.

References

Last revised on April 8, 2021 at 09:16:29. See the history of this page for a list of all contributions to it.