Conformally Einstein product spaces

TL;DR

This paper classifies conformally Einstein product spaces, especially in low dimensions, providing explicit examples and advancing understanding of their geometric structure and classification.

Contribution

It offers a comprehensive classification of conformally Einstein product spaces, including explicit solutions and special cases in low dimensions.

Findings

Complete classification of conformally Einstein product spaces in various dimensions.

Explicit examples of Einstein warped products with 1D and 2D factors.

Identification of special families of metrics with conformal gradient fields.

Abstract

We study pseudo-Riemannian Einstein manifolds which are conformally equivalent with a metric product of two pseudo-Riemannian manifolds. Particularly interesting is the case where one of these manifolds is 1-dimensional and the case where the conformal factor depends on both manifolds simultaneously. If both factors are at least 3-dimensional then the latter case reduces to the product of two Einstein spaces, each of the special type admitting a non-trivial conformal gradient field. These are completely classified. If each factor is 2-dimensional, there is a special family of examples of non-constant curvature (called extremal metrics by Calabi), where in each factor the gradient of the Gaussian curvature is a conformal vector field. Then the metric of the 2-manifold is a warped product where the warping function is the first derivative of the Gaussian curvature. Moreover we find explicit examples of Einstein warped products with a 1-dimensional fibre and such with a 2-dimensional base. Therefore in the 4-dimensional case our Main Theorem points towards a local classification of conformally Einstein products. Finally we prove an assertion in the book by A.Besse on complete Einstein warped products with a 2-dimensional base. All solutions can be explicitly written in terms of integrals of elementary functions.