Study of P -curvature tensor and other related tensors Ganesh Prasad Pokhariyal Abstract. In this paper the relationships between W 2 , P and other re- lated tensors have been obtained and corresponding propositions are made. Further, the condition for P -curvature tensor to satisfy the Bianchi differ- ential identity has been established. M.S.C. 2010: 53C50, 83C35. Key words: W 2 −curvature tensor; P -curvature tensor; Codazzi tensor; Bianchi identity. 1 Introduction The W 2 curvature tensor defined by [4] has been widely studied in differential geome- try as well as in the space time of general relativity. [3] have studied it in P -Sasakian manifold; [5] studied it for Sasakian manifold. [7] have introduced the notion of weekly W 2 -symmetric manifolds and studied their properties. [10] have studied this tensor in Kernmotsu manifolds, while [8] considered N (k)−quasi Einstein manifolds satisfy- ing the conditionsR(ξ,X).W 2 =0. Further [9] have studied Lorentzian Para-Sasakian manifold satisfying some conditions on W 2 -curvature tensor. [1] have studied space times satisfying Einstein field equations with vanishing of W 2 -curvature as well as existence of killing and conformal killing vector fields. Further, the vanishing and divergence of W 2 -tensor have also been studied in perfect fluid space-times. The P -curvature tensor has been defined by breaking the W 2 -curvature tensor in skew -symmetric part and some of its properties have been studied [4]. Further, W 2 - curvature tensor was shown to extend Pirani formulation of gravitational waves to Einstein space ([6]). Consider an n-dimensional space V n in which the tensors: (1.1) C (X,Y,Z,T )= R(X,Y,Z,T ) − (R/n(n − 1))[g(X, T )g(Y,Z ) − g(Y,T )g(X, Z )] L(X,Y,Z,T )=R(X,Y,Z,T ) - (1/n - 2)[g(Y,Z)Ric(X, T ) - g(X, Z)Ric(Y,T )+ g(X, T )Ric(Y,Z) - g(Y,T )Ric(X, Z)] (1.2) V (X,Y,Z,T )=R(X,Y,Z,T ) − (1/n − 2)[g(X, T )Ric(Y,Z ) − g(Y,T )Ric(X, Z )+ g(Y,Z )Ric(X, T ) − g(X, Z )Ric(Y,T )] + R/(n − 1)(n − 2)[g(X, T )g(Y,Z ) − g(Y,T )g(X, Z )] (1.3) Differential Geometry - Dynamical Systems, Vol.22, 2020, pp. 202-207. c ⃝ Balkan Society of Geometers, Geometry Balkan Press 2020.