A family of special case of sequential warped product manifolds with semi-Riemannian Einstein metrics

TL;DR

This paper derives formulas for special semi-Riemannian warped product manifolds to be Einstein and explores solutions when base manifolds are conformal to pseudo-Euclidean spaces, including invariant solutions with positive Ricci curvature.

Contribution

It introduces a family of Einstein metrics on sequential warped product manifolds with conformal base spaces, expanding understanding of Einstein structures in semi-Riemannian geometry.

Findings

Derived general formulas for Einstein conditions in special warped product configurations.

Proved existence of invariant solutions for positive Ricci curvature case.

Identified conditions under which base manifolds are conformal to pseudo-Euclidean spaces.

Abstract

We derive the general formulas for a special configuration of the sequential warped product semi-Riemannian manifold to be Einstein, where the base-manifold is the product of two manifolds both equipped with a conformal metrics. Subsequently we study the case in which these two manifolds are conformal to a $n_1$-dimensional and $n_2$-dimensional pseudo-Euclidean space, respectively. For the latter case, we prove the existence of a family of solutions that are invariant under the action of a $(n_1-1)$-dimensional group of transformations to the case of positive constant Ricci curvature ($\lambda>0$).