Riemannian counterparts to Lorentzian space forms

On a smooth $n$-manifold $M$ with $n \geq 3$, we study pairs $(g,T)$ consisting of a Riemannian metric $g$ and a unit length closed vector field $T$. Motivated by how Ricci solitons generalize Einstein metrics via a distinguished vector field, we propose to generalize space forms by considering those pairs $(g,T)$ whose corresponding Lorentzian metric $g_{\scriptscriptstyle L} = g - 2T^{\flat} \otimes T^{\flat}$ has constant curvature. We show by examples that such pairs exist when $M$ is noncompact, and that complete metrics exist among them. When $M$ is compact, however, the situation is more rigid. In the compact setting, we prove that the only pairs $(g,T)$ whose corresponding Lorentzian metric $g_{\scriptscriptstyle L}$ is a space form are those where $(M,g)$ is flat and its universal covering splits isometrically as a product $\mathbb{R} \times N$. The nonexistence of compact Lorentzian spherical space forms plays a key role in our proof.