Mathematics
On the Einstein condition for Lorentzian 3-manifolds
Document Type
Article
Abstract
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=fg+(f−λ)T♭⊗T♭, for any unit timelike vector field T, any positive constant λ, and any nontrivial function f that never takes the value λ. (Observe that this reduces to the positive Einstein case when f=λ.) We show that there is no such obstruction if λ is negative. Finally, the “borderline” case λ=0 is also examined: we show that if λ=0, then (M,g) must be isometric to (S1×N,−dt2⊕h) with (N,h) a Riemannian manifold.
Publication Title
Journal of Mathematical Analysis and Applications
Publication Date
5-15-2021
Volume
497
Issue
2
ISSN
0022-247X
DOI
10.1016/j.jmaa.2020.124892
Keywords
dimension three, Einstein metrics, Lorentzian spherical space forms
Repository Citation
Aazami, Amir Babak, "On the Einstein condition for Lorentzian 3-manifolds" (2021). Mathematics. 23.
https://commons.clarku.edu/mathematics/23
