Complex Manifolds (Jan 2019)

On curvature tensors of Norden and metallic pseudo-Riemannian manifolds

  • Blaga Adara M.,
  • Nannicini Antonella

DOI
https://doi.org/10.1515/coma-2019-0008
Journal volume & issue
Vol. 6, no. 1
pp. 150 – 159

Abstract

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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 {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.

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