Conformally recurrent space-times admitting a proper conformal vector field
Commun. Korean Math. Soc. 2014 Vol. 29, No. 2, 319-329
https://doi.org/10.4134/CKMS.2014.29.2.319
Printed April 1, 2014
Uday Chand De and Carlo Alberto Mantica
University of Calcutta, Unicersita Degli Studi Di Milano
Abstract : In this paper we study the properties of conformally recurrent pseudo Riemannian manifolds admitting a proper conformal vector field with respect to the scalar field $\sigma ,$ focusing particularly on the 4-dimensional Lorentzian case. Some general properties already proven by one of the present authors for pseudo conformally symmetric manifolds endowed with a conformal vector field are proven also in the case, and some new others are stated. Moreover interesting results are pointed out; for example, it is proven that the Ricci tensor under certain conditions is {\it Weyl compatible}: this notion was recently introduced and investigated by one of the present authors. Further we study conformally recurrent 4-dimensional Lorentzian manifolds (space-times) admitting a conformal vector field: it is proven that the covector $\sigma _{j}$ is null and unique up to scaling; moreover it is shown that the same vector is an eigenvector of the Ricci tensor. Finally, it is stated that such space-time is of Petrov type N with respect to $\sigma _{j}.$
Keywords : conformally recurrent space-times, proper conformal vector fields, pseudo-Riemannian manifolds, Weyl compatible tensors, Petrov types, Lorentzian metrics
MSC numbers : Primary 53B20, 53C50; Secondary 83C20
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