Alternative Equations for Horizontal Lifts of the Metallic Structures from Manifold onto Tangent Bundle

Abstract

We explore “the horizontal lift” of the structure J satisfying J2−αJ−βI=0 and establish that it as a kind of metallic structure. An analysis of Nijenhuis tensor of metallic structure JH is presented, and a new tensor field J˜ of 1,1-type is introduced and demonstrated to be metallic structure. Some results on the Nijenhuis tensor and the Lie derivative of J˜ in TM are proved and explicit examples are given. Moreover, the metallic structure J˜ endowed with projection operators l˜ and m˜ in TM is studied.