# nLab invariant differential form

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

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

(1)$(L_g)^* \omega = \omega$

along the left multiplication action

$\array{ L_g \colon & G &\longrightarrow& G \\ & x &\mapsto& g \cdot x }$

Analogously a form is right invariant if it is invariant under the pullback by right translations $R_g$.

More generally, given a differentiable (e.g. smooth) group action of $G$ on a differentiable (e.g. smooth) manifold $M$

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

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

$\rho(g)^\ast(\omega) \;=\; \omega \,.$

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

### Invariant vector field

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