Neutral bi-Hermitian Gray surfaces

Abstract

The aim of this paper is to give examples of compact neutral 4-manifolds $(M,g)$ whose Ricci tensor $\rho$ satisfies the relation ${\nabla }_X\rho (X,X)=\frac{1}{3}X\tau g(X,X)$. We present also a family of new Einstein bi-Hermitian neutral metrics on ruled surfaces of genus $g>1$.

12 pages