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On a class of generalized quasi-Einstein manifolds with applications to relativity. (English) Zbl 1366.53033

A Riemannian or pseudo-Riemannian manifold \((M^n,g)\) of dimension \(n\) is said to be a generalized quasi-Einstein manifold, denoted by \(\mathrm G(\mathrm{QE})_n\), if its Ricci tensor \(S\) satisfies \(S(X, Y)=ag(X,Y)+bA(X)A(Y)+c[A(X)B(Y)+A(Y)B(X)]\), where \(a,b,c\) are scalar functions and \(A,B\) are 1-forms on \(M\). The aim of the present paper is to investigate the \(\mathrm G(\mathrm{QE})_n\) spaces for \(n>3\). Some properties of \(\mathrm G(\mathrm{QE})_n\) spaces are found and their relations to other structures on \((M^n,g)\) are studied. The \(\mathrm G(\mathrm{QE})_4\) spacetimes with vanishing space-matter tensor and with divergence free space-matter tensor are discussed. Two nontrivial examples of generalized quasi-Einstein spacetimes are constructed.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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