# Which differential forms commute with the curvature form?

Consider a vector bundle, $E \to M$, with connection, $\nabla$, and curvature $2$-form, $F$ on $M$. For $E$-valued differential forms on $M$, $\Omega(M, E)$, we have an exterior covariant derivative, $d_{\nabla}$ for which $d_{\nabla}^2 = F \wedge (-)$.

Question: What is the significance of the collection of forms, $$\{ \omega \in \Omega(M, E) \ | \ F \wedge \omega = 0 \}?$$

• I would say these forms annihilate, not commute. – Thomas Rot Aug 18 '17 at 13:51
• In the context where, locally, the forms take values in some lie algebra that wedge becomes a wedge-bracket and since I think of bracketing as being motivated from the commutator... hence the name. But sure. Either context could be helpful! – cheyne Aug 18 '17 at 14:03