A remark about static space times (Q2518937)

The author studies the equation \[ Ddf+(\Delta f)g-f\operatorname{Ric}=0 \] well known in general relativity, whose unknowns are the metric \(g\) and the non-identically vanishing function \(f\), on a compact manifold \(M\). Mainly, he proves the following Theorem: Let \((M,g)\) be a compact three dimensional Riemannian manifold such that \(S(g)\) has two linearly independent solutions. Then \((M,g)\) is isometric to a standard product \(S^{1}\times S^{2}\), \(S^{1}\times \mathbb{P}^{2} \mathbb{R}\) or the standard 3-sphere.