Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 15 / 5 / b

Use the sign convention in the supplied scalar Laplace-Beltrami operator identity, and let $X={\omega }^{♯}$ be the metric-dual vector field. Integrate that Bochner formula for one-forms over the closed Riemannian manifold, with its volume density $d{V}_g$. 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. HenceBoth terms are nonnegative, because $\mathrm{Ric}\ge 0$. 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, provingThus harmonic one-forms are parallel under nonnegative Ricci curvature. The same argument also gives $\mathrm{Ric}(X,X)=0$ pointwise. No orientation or connectedness assumption is needed for this step.