  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.
