= Solution
Use the sign convention in the supplied scalar <Laplace-Beltrami operator> identity, and let $X=\omega^\sharp$ be the metric-dual <vector field>. Integrate that <Bochner formula for one-forms> over the closed <Riemannian manifold>, with its volume density $dV_g$. The integral of a scalar Laplacian is zero: on an oriented manifold this is the <Riemannian divergence theorem> applied to its gradient, and the same density formula is valid without orientability. Hence
$$
0=\int_M\left(|\nabla\omega|^2+\operatorname{Ric}(X,X)\right)dV_g.
$$
Both terms are nonnegative, because $\operatorname{Ric}\ge0$. The smooth nonnegative integrand must therefore vanish everywhere: a strictly positive value would remain positive on an open neighborhood of positive volume and make its integral positive. In particular $|\nabla\omega|^2=0$ at every point, proving
$$
\boxed{\nabla\omega=0.}
$$
Thus <harmonic one-forms are parallel under nonnegative Ricci curvature>. The same argument also gives $\operatorname{Ric}(X,X)=0$ pointwise. No <orientation> or <connectedness> assumption is needed for this step.
Back to article page
