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Conformal Einstein spaces and conformally covariant operators

Alfonso García-Parrado, Jónatan Herrera, Miguel Vadillo

TL;DR

This work develops a dimension independent framework to decide when a pseudo-Riemannian manifold is conformal to an Einstein space by leveraging conformally covariant metric concomitants and the $\mathcal{C}$-connection. It introduces conformally covariant differentiation operators that act on scalars and tensors, along with singular counterparts $\mathcal{C}^\xi$ when the Weyl endomorphism is not invertible, enabling a complete characterization via $\mathcal{C}$-Ricci data. The main results give necessary and sufficient conditions for conformal Einstein-ness in both invertible and singular Weyl scenarios, and are illustrated with conformally flat, Robinson-Trautmann, and pp-wave examples. The approach yields algorithmic criteria and a unified higher dimensional treatment that clarifies obstructions and constructive conformal factors. Overall, the paper advances conformal geometry in arbitrary dimensions by providing computational tools and explicit exemplars for conformal Einstein spaces.

Abstract

In this article we give general neccessary and sufficient conditions to ensure that a pseudo-Riemannian manifold is conformal to an Einstein space. These conditions are algorithmic in \emph{the metric tensor} whenever the Weyl endomorphism is invertible. Our conditions depend in an essential manner on the $\mathcal{C}$-connection. We also show how to construct \emph{conformally covariant, pseudo-differential} operators which has an independent interest.

Conformal Einstein spaces and conformally covariant operators

TL;DR

This work develops a dimension independent framework to decide when a pseudo-Riemannian manifold is conformal to an Einstein space by leveraging conformally covariant metric concomitants and the -connection. It introduces conformally covariant differentiation operators that act on scalars and tensors, along with singular counterparts when the Weyl endomorphism is not invertible, enabling a complete characterization via -Ricci data. The main results give necessary and sufficient conditions for conformal Einstein-ness in both invertible and singular Weyl scenarios, and are illustrated with conformally flat, Robinson-Trautmann, and pp-wave examples. The approach yields algorithmic criteria and a unified higher dimensional treatment that clarifies obstructions and constructive conformal factors. Overall, the paper advances conformal geometry in arbitrary dimensions by providing computational tools and explicit exemplars for conformal Einstein spaces.

Abstract

In this article we give general neccessary and sufficient conditions to ensure that a pseudo-Riemannian manifold is conformal to an Einstein space. These conditions are algorithmic in \emph{the metric tensor} whenever the Weyl endomorphism is invertible. Our conditions depend in an essential manner on the -connection. We also show how to construct \emph{conformally covariant, pseudo-differential} operators which has an independent interest.
Paper Structure (13 sections, 8 theorems, 107 equations)

This paper contains 13 sections, 8 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Notions of conformal geometry
  3. Conformally covariant differentiation
  4. The fundamental lemma
  5. Conformal differentiation of conformally covariant metric concomitants
  6. The $\mathcal{C}$ and $\mathcal{C}^\xi$-connections
  7. Characterization of conformal Einstein spaces
  8. Examples
  9. Conformally flat spaces
  10. The Robinson--Trautmann class of solutions
  11. Plane fronted waves
  12. Conclusions
  13. The Moore-Penrose pseudo-inverse

Key Result

Lemma 3.1

Let where $\xi^p{}_{[mn]}=\xi^p{}_{mn}$, $\tilde{\xi}^p{}_{[mn]}=\tilde{\xi}^p{}_{mn}$ are some tensors and $\tilde{\xi}^p{}_{mn}$ is arbitrary. If the Weyl endomorphism is invertible then $\Upsilon_{a}$ defined in upsilon definition fulfills the relation Otherwise the relation is

Theorems & Definitions (22)

  • Definition 2.1
  • Definition 2.2
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Theorem 3.5
  • proof
  • ...and 12 more