Riemannian counterparts to Lorentzian space forms

TL;DR

This paper introduces a new class of geometric structures on manifolds, generalizing space forms via pairs of Riemannian metrics and vector fields, and explores their existence and properties in both compact and noncompact cases.

Contribution

It defines Riemannian counterparts to Lorentzian space forms using pairs (g,T) and analyzes their existence, especially highlighting the rigidity in the compact case.

Findings

Existence of such pairs on noncompact manifolds with complete metrics.

Rigidity results showing only flat cases in the compact setting.

Nonexistence of compact Lorentzian spherical space forms.

Abstract

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.