Irreducible holonomy in dimension at least two has no parallel one-form
= Irreducible holonomy in dimension at least two has no parallel one-form
A nonzero parallel one-form determines a nonzero parallel vector by metric duality. Its value is fixed by the full <Riemannian holonomy group>, so its span is an invariant line. This contradicts irreducibility when $\dim M\ge2$. The dimension condition matters: the standard circle has trivial but irreducible one-dimensional holonomy representation and a nonzero parallel one-form.