On W 2 - Curvature Tensor Nk-quasi Einstein Manifolds

Abstract

We consider N(k)-quasi Einstein manifolds satisfying the conditions R(ξ,X). 2 W = 0, 2 W (ξ,X).S = 0, P(ξ,X). 2 W = 0, where 2 W and P denote the 2 W-curvature tensor and projective curvature tensor, respectively. M.S.C: 53C25.

The $W$-curvature tensor on relativistic space-times

“…They investigated their geometrical properties and physical significance. These tensors have been recently studied in different ambient spaces [1,4,5,11,17,18,20]. However, we noticed that a little attention is paid to the W * 3 −curvature tensor.…”

Curvature tensor for the spacetime of general relativity

“…On the other hand, in terms of W 2 -tensor, Shaikh et al [10] have introduced the notion of weekly W 2 -symmetric manifolds and studied their properties along with several non-trivial examples. The role of W 2 -tensor in the study of Kenmotsu manifolds has been investigated by Yildiz and De [14] while N(k)-quasi Einstein manifolds satisfying the conditions R(ξ, X).W 2 = 0 have been considered by Taleshian and Hosseinzadeh [12]. Most recently, Venkatesha et al [13] have studied Lorentzian para-Sasakian manifolds satisfying certain conditions on W 2 -curvature tensor.…”

EXISTENCE OF N(k)-QUASI EINSTEIN MANIFOLDS

“…Again, in 2008,Özgür [12] studied the deviation conditions R(ξ, X).P = 0, P (ξ, X).S = 0 and P (ξ, X).P = 0 for an N (k)−quasi Einstein manifold, where P denotes the projective curvature tensor and some physical examples of N (k)−quasi Einstein manifolds are given. In 2010, Singh, Pandey and Gautam [13], Taleshian and Hosseinzadeh [17], Dwivedi [24] have studied the N (k)−quasi Einstein manifolds with the deviation conditions R(ξ, X).P = 0, R(ξ, X).W 2 = 0, W 2 (ξ, X).S = 0 andP (ξ, X).P = 0, whereP and W 2 denote the pseudo projective curvature and W 2 −projective curvature tensors, respectively. Several geometrical properties of N (k)−quasi Einstein manifolds have studied by Yildiz, De and Cetinkaya [15], Taleshian and Hosseinzadeh ( [16], [23]), De, De and Gazi [14], Yang and Xu [22] and others.…”