# nLab invariant differential form

Invariant differential forms and vector fields

### 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

tangent cohesion

differential cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Invariant differential forms and vector fields

## Definition

Let $M$ be a differential manifold with differentiable left action of Lie group $G$, $G\times M\to M$ (respectively right action $M\times G\to M$). For example, the multiplication map of $G$ on itself. Then we define the left translations $L_g : m\mapsto g m$ (resp. right translations $R_g: m\mapsto m g$) for every $g\in G$, which are both diffeomorphisms of $M$.

A differential form on a Lie group $\omega \in \Omega^1(G)$ is called left invariant if for every $g \in G$ it is invariant under the pullback by the translation $L_g$

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

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