Riemannian counterparts to Lorentzian space forms
On a smooth -manifold with , we study pairs consisting of a Riemannian metric and a unit length closed vector field . Motivated by how Ricci solitons generalize Einstein metrics via a distinguished vector field, we propose to generalize space forms by considering those pairs whose corresponding Lorentzian metric has constant curvature. We show by examples that such pairs exist when is noncompact, and that complete metrics exist among them. When is compact, however, the situation is more rigid. In the compact setting, we prove that the only pairs whose corresponding Lorentzian metric is a space form are those where is flat and its universal covering splits isometrically as a product . The nonexistence of compact Lorentzian spherical space forms plays a key role in our proof.