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Keywords = almost quadratic ϕ-structure

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16 pages, 323 KiB  
Article
Metallic Structures for Tangent Bundles over Almost Quadratic ϕ-Manifolds
by Mohammad Nazrul Islam Khan, Sudhakar Kumar Chaubey, Nahid Fatima and Afifah Al Eid
Mathematics 2023, 11(22), 4683; https://doi.org/10.3390/math11224683 - 17 Nov 2023
Cited by 2 | Viewed by 1208
Abstract
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, [...] Read more.
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. Full article
(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Submanifolds)
12 pages, 288 KiB  
Article
Characterizations of the Frame Bundle Admitting Metallic Structures on Almost Quadratic ϕ-Manifolds
by Mohammad Nazrul Islam Khan, Uday Chand De and Teg Alam
Mathematics 2023, 11(14), 3097; https://doi.org/10.3390/math11143097 - 13 Jul 2023
Cited by 3 | Viewed by 1147
Abstract
In this work, we have characterized the frame bundle FM admitting metallic structures on almost quadratic ϕ-manifolds ϕ2=pϕ+qIqηζ, where p is an arbitrary constant and q is a [...] Read more.
In this work, we have characterized the frame bundle FM admitting metallic structures on almost quadratic ϕ-manifolds ϕ2=pϕ+qIqηζ, where p is an arbitrary constant and q is a nonzero constant. The complete lifts of an almost quadratic ϕ-structure to the metallic structure on FM are constructed. We also prove the existence of a metallic structure on FM with the aid of the J˜ tensor field, which we define. Results for the 2-Form and its derivative are then obtained. Additionally, we derive the expressions of the Nijenhuis tensor of a tensor field J˜ on FM. Finally, we construct an example of it to finish. Full article
(This article belongs to the Special Issue Geometry of Manifolds and Applications)
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