Palestine Journal of Mathematics Vol. 3(1) (2014) , 61–69 © Palestine Polytechnic University-PPU 2014 On some properties of W -curvature tensor Zafar Ahsan and Musavvir Ali Communicated by Jose Luis Lopez-Bonilla MSC 2010 Classifications: 53C25,53C50, 83C20, 83C50. 11R32. Keywords and phrases: curvature tensors, collineation, electromagnetic fields. Abstract. Relationship between W -curvature tensor and its divergence with that of other curvature tensors has been established. A symmetry of the spacetime named as W -collineation, has been introduced and conditions under which the spacetimes of general relativity may admit such collineations are obtained. 1 Introduction The construction of gravitational potentials satisfying Einstein’s field equations is the principal aim of all investigations in gravitational physics and this has been often been achieved by im- posing symmetries on the geometry compatible with the dynamics of the chosen distribution of matter. The geometrical symmetries of the spacetime are expressible through the vanishing of the Lie derivative of certain tensors with respect to a vector. This vector may be time-like, space-like or null. The role of symmetries in general theory of relativity has been introduced by Katzin, Levine and Davis in a series of papers ([11] - [13]). These symmetries, also known as collineations, were further studied by Ahsan ([1] - [5]), Ahsan and Ali [7] and Ahsan and Husain [9]. In the differential geometry of certain F -structures, W -curvature tensor has been studied by a number of workers especially by Pokhriyal [16] for a Sasakian manifold; while for a P-Sasakian manifold Matsumoto et al [14] have studied this tensor. Shaikh et al [18] have introduced the no- tion of weekly W 2 -symmetric manifolds in terms of W 2 -tensor and studied their properties along with numerous non-trivial examples. The role of W 2 -tensor in the study of Kenmotsu manifolds has been investigated by Yildiz and De [23] while N(k)-quasi Einstein manifolds satisfying the conditions R(ξ,X).W 2 = 0 have been considered by Taleshian and Hosseinzadeh [20]. Most re- cently, Venkatesha et al [21] have studied Lorentzian para-Sasakian manifolds satisfying certain conditions on W -curvature tensor. Motivated by the all important role of W -curvature tensor in the study of certain differential geometric structures, Ahsan et al. [8] have made a detailed study of this tensor on the spacetime of general relativity. The purpose of this paper is to develop the relationships between the divergences of W , pro- jective, conformal, conharmonic and concircular curvature tensors and to introduce a symmetry property of spacetime of general relativity, known as W -collineation, defined through the van- ishing of Lie derivative of W -curvature tensor with respect to a vector field. The divergences are given in Section 3; while in Section 4, we have discussed W -collineation with some results and the cases of non-null and null electromagnetic fields are discussed in this context. Finally, in Section 5 summary of the work is given. 2 Preliminaries So far more than twenty six different types of collineations have been studied and the litera-