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Weakly \(Z\)-symmetric manifolds. (English) Zbl 1289.53040

On a Riemannian manifold \((M,g)\), the authors define a \((0,2)\)-tensor field \(Z\), by the formula \[ Z_{ij} = (\mathrm{Ric})_{ij} + \Phi g \] where \(\Phi\) is an arbitrary function on \(M\) and Ric is the Ricci tensor field. If, moreover, there exist three 1-forms \(A\), \(B\), \(D\) such that \[ \nabla_k Z_{jl} = A_k Z_{jl} + B_j Z_{kl} + D_l Z_{kj} \] then the manifold \((M,g)\) is called weakly \(Z\)-symmetric (abreviated WZS). The WZS manifolds generalize simultaneously the weakly-, the pseudo- and the pseudo projective Ricci symmetric manifolds.
On WZS manifolds, the authors study the existence of proper concircular vectors; in some particular cases, the form of the Ricci tensor field and the local form of the metric are determined.

MSC:

53B20 Local Riemannian geometry
53B21 Methods of local Riemannian geometry
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References:

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