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Curvature properties of pseudosymmetry type of some 2-quasi-Einstein manifolds

Ryszard Deszcz, Małgorzata Głogowska, Jan Jełowicki, Miroslava Petrović-Torgašev, Georges Zafindratafa

Abstract

Let (M,g) be a 2-quasi-Einstein non-conformally flat semi-Riemannian manifold of dimension > 3. We prove that if its Riemann-Christoffel curvature tensor R is a linear combination of some Kulkarni-Nomizu tensors formed by the metric tensor g, the Ricci tensor S and its square S^2, then some pseudosymmetry type curvature conditions are satisfied. Certain non-conformally flat warped product manifolds with 2-dimensional base, and in particular some spacetimes, are such 2-quasi Einstein manifolds.

Curvature properties of pseudosymmetry type of some 2-quasi-Einstein manifolds

Abstract

Let (M,g) be a 2-quasi-Einstein non-conformally flat semi-Riemannian manifold of dimension > 3. We prove that if its Riemann-Christoffel curvature tensor R is a linear combination of some Kulkarni-Nomizu tensors formed by the metric tensor g, the Ricci tensor S and its square S^2, then some pseudosymmetry type curvature conditions are satisfied. Certain non-conformally flat warped product manifolds with 2-dimensional base, and in particular some spacetimes, are such 2-quasi Einstein manifolds.
Paper Structure (6 sections, 18 theorems, 227 equations)

This paper contains 6 sections, 18 theorems, 227 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Symmetric (0,2)-tensors
  4. Hypersurfaces satisfying pseudosymmetry type curvature conditions
  5. Some special generalized curvature tensors
  6. Warped product manifolds with 2-dimensional base manifold

Key Result

Proposition 2.1

Let $B$ be a generalized curvature tensor on a semi-Riemannian manifold $(M,g)$, $\dim M = n \geq 4$. If at every point $x \in \mathcal{U}_{\mathrm{Ric} (B)} \cap \mathcal{U}_{ \mathrm{Weyl} (B)} \subset M$ the local components of the tensor $Weyl(B)$, which may not vanish identically, are the follo then on $\mathcal{U}_{\mathrm{Ric} (B)} \cap \mathcal{U}_{ \mathrm{Weyl} (B)}$, where $a,b,c,d \in

Theorems & Definitions (18)

  • Proposition 2.1
  • Proposition 2.2
  • Corollary 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Lemma 3.1
  • Lemma 3.2
  • ...and 8 more