Quasi-Einstein shearfree spacetimes lifted from Sasakian manifolds

TL;DR

This paper demonstrates that certain smooth Sasakian manifolds can be lifted to 4D quasi-Einstein shearfree spacetimes of Petrov type II or D, extending previous results to a broader class of manifolds.

Contribution

It establishes new conditions under which Sasakian manifolds admit lifts to specific shearfree Einstein spacetimes, broadening the scope beyond real-analytic CR manifolds.

Findings

Sasakian manifolds admit lifts to quasi-Einstein shearfree spacetimes of Petrov type II or D

All tubular CR manifolds satisfy the lift condition

Sasakian manifolds with Kähler-Einstein base and non-zero Einstein constant can be lifted to shearfree Einstein metrics

Abstract

In this article we prove that a certain class of {\it smooth} Sasakian manifolds admits lifts to 4-dimensional quasi-Einstein shearfree spacetimes of Petrov type II or D. This is related to an analogous result by Hill, Lewandowski and Nurowski \cite{HLN} for general {\it real-analytic} CR manifolds. In particular, this holds for all tubular CR manifolds. Furthermore, we show that any Sasakian manifold with underlying K\"ahler-Einstein manifold with non-zero Einstein constant has a lift to a shearfree Einstein metric of Petrov type II or D.