Abstract
It is known that the difference tensor \(R \cdot C - C \cdot R\) and the Tachibana tensor \(Q(S,C)\) of any semi-Riemannian Einstein manifold \((M,g)\) of dimension \(n \ge 4\) are linearly dependent at every point of \(M\). More precisely \(R \cdot C - C \cdot R = (1/(n-1))\, Q(S,C)\) holds on \(M\). In the paper we show that there are quasi-Einstein, as well as non-quasi-Einstein semi-Riemannian manifolds for which the above mentioned tensors are linearly dependent. For instance, we prove that every non-locally symmetric and non-conformally flat manifold with parallel Weyl tensor (essentially conformally symmetric manifold) satisfies \(R \cdot C = C \cdot R = Q(S,C) = 0\). Manifolds with parallel Weyl tensor having Ricci tensor of rank two form a subclass of the class of Roter type manifolds. Therefore we also investigate Roter type manifolds for which the tensors \(R \cdot C - C \cdot R\) and \(Q(S,C)\) are linearly dependent. We determine necessary and sufficient conditions for a Roter type manifold to be a manifold having that property.
Highlights
Let ∇, R, S, κ and C be the Levi-Civita connection, the Riemann–Christoffel curvature tensor, the Ricci tensor, the scalar curvature tensor and the Weyl conformal curvature tensor of a semi-Riemannian manifold (M, g), n = dim M ≥ 2, respectively. It is well-known that the manifold (M, g), n ≥ 3, is said to be an Einstein manifold ([1])
We denote by US the set of all points of (M, g) at which S is not proportional to g, i.e., US
The remarks above lead to the problem of investigation of curvature properties of nonEinstein and non-conformally flat semi-Riemannian manifolds (M, g), n ≥ 4, satisfying at every point of M the curvature condition, of the following form: the difference tensor R · C − C · R is proportional to Q(g, R), Q(S, R), Q(g, C) and Q(S, C). Such conditions are strongly related to some pseudosymmetry type curvature conditions, see, e.g., [2] and references therein
Summary
The remarks above lead to the problem of investigation of curvature properties of nonEinstein and non-conformally flat semi-Riemannian manifolds (M, g), n ≥ 4, satisfying at every point of M the curvature condition, of the following form: the difference tensor R · C − C · R is proportional to Q(g, R), Q(S, R), Q(g, C) and Q(S, C). Such conditions are strongly related to some pseudosymmetry type curvature conditions, see, e.g., [2] and references therein. We show (Example 5.4) that under some conditions the Cartesian product of two semi-Riemannian spaces of constant curvature satisfies assumptions of Theorem 5.3
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