On curvature tensors of Norden and metallic pseudo-Riemannian manifolds (Q2311393)

Let \((M,g)\) be a pseudo-Riemannian manifold. A metallic pseudo-Riemannian structure \(J\) on \(M\) is a \(g\)-symmetric \((1,1)\)-tensor field on \(M\) such that \(J^2=pJ+qI\), for some \(p,q\in\mathbb R\). In particular, for \(p=0\) and \(q=-1\), \(J\) is a Norden structure on \(M\). The present paper is devoted to the study of some differential geometric properties of Norden and, more generally, metallic pseudo-Riemannian manifolds. In particular, the authors define nearly locally metallic and nearly Kähler-Norden manifolds and introduce the notion of \(J\)-sectional and \(J\)-bisectional curvature of a metallic pseudo-Riemannian manifold \((M,J,g)\) and study their properties. It is proved that under certain assumptions, if the manifold is locally metallic, then the Riemannian curvature tensor vanishes.