Abstract
AbstractWe 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 {Ja,b}a,b∈ℝ and show that for a ≠ 0, the J-sectional and J-bisectional curvatures of M coincide with the Ja,b-sectional and Ja,b-bisectional curvatures, respectively. We also give examples of Norden and metallic structures on ℝ2n.
Highlights
Let (M, g) be a pseudo-Riemannian manifold
We study some properties of curvature tensors of Norden and, more generally, metallic pseudoRiemannian manifolds
We prove that under certain assumptions, if the manifold is locally metallic, the Riemann curvature tensor vanishes
Summary
Let (M, g) be a pseudo-Riemannian manifold. A metallic pseudo-Riemannian structure J on M is a gsymmetric ( , )-tensor eld on M such that J = pJ + qI, for some p and q real numbers, [1], [7]. In the case of Norden manifolds we give a formula that express the sectional curvature with respect to the J-sectional curvature and some other terms (Proposition 4.10). This formula is the analogue of Vanhecke’s formula for almost hermitian manifolds, [14], in the Kähler-Norden case it is not possible to simplify the other terms like in [14]. Using a Norden structure (J, g) on M, we consider a family of metallic pseudo-Riemannian structures {Ja,b}a,b∈R and show that for a ≠ , the J-sectional and J-bisectional curvatures of M coincide with the Ja,b-sectional and Ja,b-bisectional curvatures, respectively. We give examples of Norden and metallic structures on R n and describe the geometrical meaning of the sign of p + q
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