The General F_(a,λ) (K,T)- Structure Satisfying aF^K+λ^r F^T=0 on M^n and Lifts Problems Associated with the Structure

Abstract

In this paper our aim is to get the general structure equation which covers previously acquired structures. In this context this paper consists of three main sections. In the first section, we define the general F_(a,λ) (K,T)-structure satisfying aF^K+λ^r F^T=0,(F≠0,K≥3,T≥1 and (K≥T),a and λ are non zero complex numbers, r some finite integer) on manifold M^n and studied to give some special examples. The second part, we find the integrability conditions by calculating Nijenhuis tensors of the horizontal lifts of the general F_(a,λ) (K,T)-structure satisfying aF^K+λ^r F^T=0. Later, we get the results of Tachibana operators applied to vector and covector fields according to the horizontal lifts of the general F_(a,λ) (K,T)-structure in cotangent bundle T^* (M^n). In addition, we have studied to show the purity conditions of Sasakian metric with respect to the horizontal lifts of the structure. In the final section, all results obtained in the second section were investigated according to the complete and horizontal lifts of the general F_(a,λ) (K,T)-structure on tangent bundle T(M^n).