Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On a certain class of para-Hermite Einstein spaces

Adam Chudecki

Abstract

A special class of (complex) para-Hermite Einstein spaces is analyzed. It is well-known that the self-dual Weyl tensor in para-Hermite Einstein spaces is of the Petrov-Penrose type [D]. In what follows we assume that the anti-self-dual Weyl tensor is algebraically degenerate. It is equivalent to the existence of an anti-self-dual congruence of null strings which is assumed not to be parallely propagated. Hence, spaces analyzed here are not Walker spaces. A classification of such spaces is given and the explicit metrics are found.

On a certain class of para-Hermite Einstein spaces

Abstract

A special class of (complex) para-Hermite Einstein spaces is analyzed. It is well-known that the self-dual Weyl tensor in para-Hermite Einstein spaces is of the Petrov-Penrose type [D]. In what follows we assume that the anti-self-dual Weyl tensor is algebraically degenerate. It is equivalent to the existence of an anti-self-dual congruence of null strings which is assumed not to be parallely propagated. Hence, spaces analyzed here are not Walker spaces. A classification of such spaces is given and the explicit metrics are found.

Paper Structure

This paper contains 21 sections, 9 theorems, 105 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Hyperheavenly spaces
  3. Hyperheavenly spaces of the types $[\textrm{D}]^{ee} \otimes [\textrm{any}]$
  4. The metric
  5. Gauge freedom
  6. Congruences of SD null strings
  7. Hyperheavenly spaces of the types $[\textrm{D}]^{ee} \otimes [\textrm{deg}]^{e}$
  8. Congruence $\mathcal{C}_{m^{\dot{A}}}$
  9. Congruences of null geodesics
  10. Hyperheavenly spaces of the types $\{ [\textrm{D}]^{ee} \otimes [\textrm{deg}]^{e} ,[+-,??] \}$
  11. ASD conformal curvature
  12. Symmetries
  13. Types $[\textrm{D}]^{ee} \otimes [\textrm{II}]^{e}$
  14. Type $\{ [\textrm{D}]^{ee} \otimes [\textrm{II}]^{e}, [+-,+-] \}$
  15. General case
  16. ...and 6 more sections

Key Result

Theorem 3.1

Let $(\mathcal{M}, ds^{2})$ be an Einstein complex space of the type $\{ [\textrm{D}]^{ee} \otimes [\textrm{II}]^{e}, [+-,+-] \}$. Then there exists a local coordinate system $(q,p,x,w)$ such that the metric takes the form where $\mu_{0}=1$, $\Lambda$ is the cosmological constant and where $M=M(q,w)$, $a=a(w)$ and $b=b(w)$ are holomorphic functions such that $M_{q} \ne 0$ and

Figures (2)

  • Figure 1: Congruences of null strings and congruences of null geodesics in algebraically degenerate para-Hermite Einstein spaces.
  • Figure 2: Congruences of null strings and congruences of null geodesics in type-[D] para-Hermite Einstein spaces.

Theorems & Definitions (24)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 3.1
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 14 more