Non-diagonal four-dimensional cohomogeneity-one Einstein metrics in various signatures

TL;DR

This paper explores non-diagonal four-dimensional cohomogeneity-one Einstein metrics, providing new explicit solutions, including neutral signature Bianchi type VIII metrics with self-dual Weyl tensor, expanding the known class of Einstein solutions.

Contribution

It explicitly constructs new non-diagonal Einstein metrics in four dimensions, especially in neutral signature, and identifies cases that exhaust possible solutions.

Findings

Constructed new neutral signature Bianchi type VIII Einstein metrics.

Found solutions with self-dual Weyl tensor.

Identified cases that exhaust possibilities for non-diagonal metrics.

Abstract

Most known four-dimensional cohomogeneity-one Einstein metrics are diagonal in the basis defined by the left-invariant one-forms, though some essentially non-diagonal ones are known. We consider the problem of explicitly seeking non-diagonal Einstein metrics, and we find solutions which in some cases exhaust the possibilities. In particular we construct new examples of neutral signature non--diagonal Bianchi type VIII Einstein metrics with self--dual Weyl tensor.