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Quasi-Einstein shearfree spacetimes lifted from Sasakian manifolds

Masoud Ganji, Gerd Schmalz, Daniel Sykes

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 Kähler-Einstein manifold with non-zero Einstein constant has a lift to a shearfree Einstein metric of Petrov type II or D.

Quasi-Einstein shearfree spacetimes lifted from Sasakian manifolds

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 Kähler-Einstein manifold with non-zero Einstein constant has a lift to a shearfree Einstein metric of Petrov type II or D.

Paper Structure

This paper contains 8 sections, 7 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. CR manifolds and Sasakian manifolds
  3. Shearfree congruences of null geodesics and CR geometry
  4. Quasi-Einstein Lorentzian manifolds
  5. Proof of Theorem \ref{['sasaki-theoint']}
  6. Proof of Theorem \ref{['tubular-theo']}
  7. Proof of Theorem \ref{['ESAKint']}
  8. Examples

Key Result

Theorem 1.1

Let $\left(M,D,J\right)$ be a 3-dimensional Sasakian manifold given by a defining equation $v = F(z,\bar{z})$, where $F\colon \mathbb C \to \mathbb R$ is such that the limits exist, and let $\Lambda$ be an arbitrary positive constant. Then, there exists a 4-dimensional Lorentzian manifold $(\mathscr M = M \times \mathbb{R}, g)$, which is a lift of $M$, such that $g$ is of Petrov type II or D, and

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • proof
  • ...and 3 more