Metallic structures on tangent bundle (original) (raw)
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In this paper, we study the natural lift types as complete and horizontal of a metallic structures on the tangent bundle of a Riemannian manifold and find some interesting results about structures, metric, distribution and the relationship between them. In the end, we introduce general natural lifts of a metallic structure and a semi-Riemannian manifold on the tangent bundle and get the conditions under which the tangent bundle endowed with such a metallic structure and a general natural lifted metric is a Riemannian manifold.
Journal of Mathematics
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
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This paper aims to explore the metallic structure J2=pJ+qI, where p and q are natural numbers, using complete and horizontal lifts on the tangent bundle TM over almost quadratic ϕ-structures (briefly, (ϕ,ξ,η)). Tensor fields F˜ and F* are defined on TM, and it is shown that they are metallic structures over (ϕ,ξ,η). Next, the fundamental 2-form Ω and its derivative dΩ, with the help of complete lift on TM over (ϕ,ξ,η), are evaluated. Furthermore, the integrability conditions and expressions of the Lie derivative of metallic structures F˜ and F* are determined using complete and horizontal lifts on TM over (ϕ,ξ,η), respectively. Finally, we prove the existence of almost quadratic ϕ-structures on TM with non-trivial examples.
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In this article, we introduce some metallic structures on the tangent bundle of a P-Sasakian manifold by complete lift, horizontal lift and vertical lift of a P-Sasakian structure (phi,eta,xi)(\phi, \eta,\xi)(phi,eta,xi) on tangent bundle. Then we investigate the integrability and parallelity of these metallic structures.
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It is well known that ?an almost complex structure? J that is J2 = ?I on the manifold M is called ?an almost Hermitian manifold? (M, J,G) if G(JX, JY) = G(X,Y) and proved that (F2M, JD,GD) is ?an almost Hermitian manifold? on the frame bundle of the second order F2M. The term ?an almost complex structure? refers to the general quadratic structure J2 = pJ + qI, where p = 0, q = ?1. However, this paper aims to study the general quadratic equation J2 = pJ + qI, where p, q are positive integers, it is named as a metallic structure. The diagonal lift of the metallic structure J on the frame bundle of the second order F2Mis studied and shows that it is also a metallic structure. The proposed theorem proves that the diagonal lift GD of a Riemannian metric G is a metallic Riemannian metric on F2M. Also, a new tensor field ?J of type (1,1) is defined on F2M and proves that it is a metallic structure. The 2-form and its derivative dF of a tensor field ?J are determined. Furthermore, the Nijen...
Horizontal lift of affinor structures and its applications
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The main purpose of the present paper is to study the horizontal lifts of tensor field of type (1, 1) (affinor field) to tensor bundle and the integrability conditions for the horizontal lifts of special types of complex and tangent structures.
Generalized metallic pseudo-Riemannian structures
arXiv: General Mathematics, 2018
We generalize the notion of metallic structure in the pseudo-Riemannian setting, define the metallic Norden structure and study its integrability. We construct a metallic natural connection recovering as particular case the Ganchev and Mihova connection, which we extend to a metallic natural connection on the generalized tangent bundle. Moreover, we construct metallic pseudo-Riemannian structures on the tangent and cotangent bundles.
On Certain Structures Defined on the Tangent Bundle
Rocky Mountain Journal of Mathematics, 2006
The differential geometry of tangent bundles was studied by several authors, for example: Davies [4], Yano and Davies [5], Dombrowski [6], Ledger and Yano [9] and Blair [1], among others. It is well known that an almost complex structure defined on a differentiable manifold M of class C ∞ can be lifted to the same type of structure on its tangent bundle T (M). However, when we consider an almost contact structure, we do not get the same type of structure on T (M). In this case we consider an odd dimensional base manifold while our tangent bundle remains to be even dimensional. The purpose of this paper is to examine certain structures on the base manifold M in relation to that of the tangent bundle T (M).