On the Einstein condition for Lorentzian 3-manifolds

It is well known that in Lorentzian geometry there are no compact spherical space forms; in dimension 3, this means there are no closed Einstein 3-manifolds with positive Einstein constant. We generalize this fact here, by proving that there are also no closed Lorentzian 3-manifolds (M, g) whose Ricci tensor satisfies Ric = f g (f - lambda)T-b circle times T-b, for any unit timelike vector field T, any positive constant lambda, and any nontrivial function f that never takes the value lambda. (Observe that this reduces to the positive Einstein case when f = lambda.) We show that there is no such obstruction if lambda is negative. Finally, the borderline case lambda = 0 is also examined: we show that if lambda = 0, then (M, g) must be isometric to (S-1 x N, -dt(2) circle plus h) with (N, h) a Riemannian manifold. (C) 2020 Elsevier Inc. All rights reserved.

Automated summary

It is proven in this article that there are no closed Lorentzian 3-manifolds whose Ricci tensor satisfies certain conditions. There is no such obstruction when lambda is negative, and when lambda is 0, the manifold must be isometric to a specific Riemannian manifold.