Basic differential form on a principal bundle

Let $\pi:P\to M$ be a principal $G$ -bundle with right action $R_g$ .

A differential form $\omega\in\Omega^k(P)$ is basic if it is both:

horizontal, and
G-invariant, meaning $(R_g)^*\omega=\omega$ for all $g\in G$ .

Theorem (descent to the base)

A form $\omega\in\Omega^k(P)$ is basic if and only if there exists a unique $\eta\in\Omega^k(M)$ such that

\pi^*\eta = \omega.

In particular, the pullback map $\pi^*:\Omega^k(M)\to \Omega^k(P)$ identifies $\Omega^k(M)$ with the subspace of basic forms on $P$ , and d preserves basic forms.

Examples

Pullbacks are basic. For any $\eta\in\Omega^k(M)$ , the form $\pi^*\eta$ is basic by construction.
Trivial bundle. For $P=M\times G$ , a form is basic exactly when it has no components along the $G$ -factor and is independent of the $G$ -coordinate; equivalently, it is pulled back from $M$ .
Hopf fibration. In the Hopf bundle $S^3\to S^2$ , the pullback of the area form on $S^2$ is a basic 2-form on $S^3$ .