Invariant differential form

Let $G$ act on a manifold $M$ by a smooth action .

A differential k-form $\omega\in \Omega^k(M)$ is $G$-invariant if for every $g\in G$,

where $\Phi_g:M\to M$ is the diffeomorphism $x\mapsto g\cdot x$.

If $G$ is connected, this is equivalent to infinitesimal invariance: for every $X$ in the Lie algebra, the Lie derivative satisfies $\mathcal{L}_{X_M}\omega=0$, where $X_M$ is the vector field generated by $X$.

Invariant forms form a subcomplex of the de Rham complex: if $\omega$ is invariant, then so is d\omega .

Examples

1. Left-invariant forms on a Lie group. For $M=G$ with the left translation action, any left-invariant 1-form (and its wedge products) is $G$-invariant.
2. Volume on the sphere. The standard volume form on $S^n$ is invariant under the natural $SO(n+1)$-action.
3. Basic forms are invariant. Any basic form on a principal bundle is invariant under the principal right action by definition.