Let $\pi:M \to B$ be a principal $G$-bundle and $\xi$ a invarint $k$-form on $M$. Does $k> dimG$ implies that $\xi$ is a basic form (pull back of a $k$-form on the base manifold $B$)?
2026-03-25 12:55:33.1774443333
Invariant forms on principal bundles
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Certainly not. Consider the wedge product of, say, the ($G$-invariant) volume form on the fiber with a basic form.