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.

Metallic Structures for Tangent Bundles over Almost Quadratic ϕ-Manifolds

“…where Υ 1 and Υ 2 are vector fields on TM. (18) and using ( 1) and ( 12), we have (18) and using ( 1) and ( 12), we have (18) and using (1) and ( 12), we have (18) and using ( 1) and ( 12), we have…”

“…Tanno [10] studied complete and vertical lifts of an almost contact structure on TM and defined a tensor field J of type (1,1) and proved that it is an almost complex structure on TM. Numerous investigators have studied various geometric structures on TM-an almost complex structure by Yano [11], paracomplex structures by Tekkoyun [12], almost r-contact structures by Das and Khan [13], and many others [14][15][16][17][18][19].…”

Tangent Bundles Endowed with Quarter-Symmetric Non-Metric Connection (QSNMC) in a Lorentzian Para-Sasakian Manifold

“…Numerous researchers have examined various connections and geometric structures on the tangent bundle like Yano and Kobayashi [2], Tani [3], Pandey and Chaturvedi [4], and Khan [5,6]. Different lifts of metallic structures to tangent bundles have been studied in [7][8][9]. Tangent bundles immersed with quarter-symmetric non-metric connections, semi-symmetric P-connections, and semi-symmetric non-metric connections on almost Hermitian manifolds, Kähler manifolds, Kenmotsu manifolds, Sasakian manifolds, para-Sasakian manifolds, Riemannian manifolds and their submanifolds, and statistical manifolds and their submanifolds have been studied in [5,[10][11][12][13][14][15][16][17][18].…”

Ricci solitons on tangent bundles with the complete lift of a projective semi-symmetric connection

“…For some significant recent papers about lifting theory on tangent bundles, we may refer to ( [1,2,6,7,8,16]).…”