Neutral bi-Hermitian Gray surfaces

TL;DR

This paper constructs examples of compact neutral 4-manifolds with specific Ricci tensor properties and introduces new Einstein bi-Hermitian neutral metrics on certain ruled surfaces, expanding the understanding of such geometric structures.

Contribution

It provides explicit examples of neutral 4-manifolds with special Ricci tensor relations and introduces new Einstein bi-Hermitian metrics on ruled surfaces of genus greater than one.

Findings

Examples of compact neutral 4-manifolds with Ricci tensor satisfying a specific relation.

New Einstein bi-Hermitian neutral metrics on ruled surfaces of genus > 1.

Enhanced understanding of geometric structures with neutral signature.

Abstract

The aim of this paper is to give examples of compact neutral 4-manifolds $(M,g)$ whose Ricci tensor $\rho$ satisfies the relation $\nabla_X\rho(X,X) =\frac13X\tau g(X,X)$. We present also a family of new Einstein bi-Hermitian neutral metrics on ruled surfaces of genus $g>1$.