Let $(M^{n},g,p)$ be a complete non-compact manifold with $\Ric_{g}\geq 0$. Let $\omega$ be a harmonic $1$-form on $(M^{n},g)$, i.e., $\Delta_{H}\omega=0$ and assume that

(4.168)

$$\lim\limits_{d_{g}(x,p)\to+\infty}|\omega(x)|=0,$$

then $\omega\equiv 0$ on $M^{n}$.