On curvature tensors of Norden and metallic pseudo-Riemannian manifolds

We study some properties of curvature tensors of Norden and, more generally, metallic pseudo-Riemannian manifolds. We introduce the notion of J -sectional and J -bisectional curvature of a metallic pseudo-Riemannian manifold ( M , J , g ) and study their properties.We prove that under certain assumptions, if the manifold is locally metallic, then the Riemann curvature tensor vanishes. Using a Norden structure ( J , g ) on M , we consider a family of metallic pseudo-Riemannian structures { J a,b } a , b ∈ℝ and show that for a ≠ 0, the J -sectional and J -bisectional curvatures of M coincide with the J a , b -sectional and J a , b -bisectional curvatures, respectively. We also give examples of Norden and metallic structures on ℝ 2 n .