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 J 2 − α J − β I = 0 and establish that it as a kind of metallic structure. An analysis of Nijenhuis tensor of metallic structure J H 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.