Bochner theorem for complete manifolds

Question

Let $(M,g)$ be a complete oriented Riemannian manifold. If $Ric ≥ 0$ on $M$, then any harmonic 1-form $\alpha$ is parallel i.e $x\to |\alpha|^2_{g,x}$ constant?

Answer 1

What about the plane? It has harmonic 1-forms, for example: real parts of holomorphic 1-forms.