# Conformally recurrent space-times admitting a proper conformal vector field

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### Additional excerpts

..., conformally recurent ([6], [20], [40], [62], [176], [223], [294], [304]), projectively recurrent [7], concircularly recurrent ([123], [131]), conharmonically recurrent [132] and quasi-conformally recurrent [65]) manifold....

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### "Conformally recurrent space-times a..." refers background in this paper

...If σj ≡ ∇jσ 6= 0 the motion is called proper, if σ is constant the vector is called homothetic [31]....

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...It is always assumed that ∇jgkl = 0 (Levi Civita connection) and that the space-matter content is described by the stress-energy tensor and related to the Ricci tensor by Einstein’s equations Rkl − R 2 gkl = κTkl, being κ = 8πG c(4) the Einstein gravitational constant (see [9], [30], [31])....

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...An n-dimensional pseudo Riemannian manifold is said to admit an infinitesimal conformal vector field (or a proper conformal motion) ξ if the Lie derivative of the metric gij along ξ satisfies the following condition (see [31] page 564 and [12]): (3) £ξgij ≡ ∇iξj +∇jξi = 2σgij , where σ is a scalar function....

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...Finally, from σCjklm = 0 and σ Rim = λσm a direct calculations brings (see also Hall’s theorem in [20] and [31]) σσRjklm = (λ− R 6 )σkσl from which it is σ[pRk]jlmσ σ = 0 and the Riemannian tensor is algebraically special....

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...When k satisfies condition b) the Weyl tensor is named algebraically special (see [12], [27], [30] and [31])....

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