Metallic Structures for Tangent Bundles over Almost Quadratic <i>ϕ</i>-Manifolds
This paper aims to explore the metallic structure <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>J</mi><mn>2</mn></msup><mo>=</mo><mi>p</...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-11-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/22/4683 |
| _version_ | 1797458527004917760 |
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| author | Mohammad Nazrul Islam Khan Sudhakar Kumar Chaubey Nahid Fatima Afifah Al Eid |
| author_facet | Mohammad Nazrul Islam Khan Sudhakar Kumar Chaubey Nahid Fatima Afifah Al Eid |
| author_sort | Mohammad Nazrul Islam Khan |
| collection | DOAJ |
| description | This paper aims to explore the metallic structure <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>J</mi><mn>2</mn></msup><mo>=</mo><mi>p</mi><mi>J</mi><mo>+</mo><mi>q</mi><mi>I</mi><mo>,</mo></mrow></semantics></math></inline-formula> where <i>p</i> and <i>q</i> are natural numbers, using complete and horizontal lifts on the tangent bundle <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mi>M</mi></mrow></semantics></math></inline-formula> over almost quadratic <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula>-structures (briefly, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>ϕ</mi><mo>,</mo><mi>ξ</mi><mo>,</mo><mi>η</mi><mo>)</mo></mrow></semantics></math></inline-formula>). Tensor fields <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>F</mi><mo>˜</mo></mover></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>F</mi><mo>*</mo></msup></semantics></math></inline-formula> are defined on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mi>M</mi></mrow></semantics></math></inline-formula>, and it is shown that they are metallic structures over <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>ϕ</mi><mo>,</mo><mi>ξ</mi><mo>,</mo><mi>η</mi><mo>)</mo></mrow></semantics></math></inline-formula>. Next, the fundamental 2-form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula> and its derivative <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>Ω</mo></mrow></semantics></math></inline-formula>, with the help of complete lift on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mi>M</mi></mrow></semantics></math></inline-formula> over <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>ϕ</mi><mo>,</mo><mi>ξ</mi><mo>,</mo><mi>η</mi><mo>)</mo></mrow></semantics></math></inline-formula>, are evaluated. Furthermore, the integrability conditions and expressions of the Lie derivative of metallic structures <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>F</mi><mo>˜</mo></mover></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>F</mi><mo>*</mo></msup></semantics></math></inline-formula> are determined using complete and horizontal lifts on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mi>M</mi></mrow></semantics></math></inline-formula> over <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>ϕ</mi><mo>,</mo><mi>ξ</mi><mo>,</mo><mi>η</mi><mo>)</mo></mrow></semantics></math></inline-formula>, respectively. Finally, we prove the existence of almost quadratic <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula>-structures on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mi>M</mi></mrow></semantics></math></inline-formula> with non-trivial examples. |
| first_indexed | 2024-03-09T16:38:25Z |
| format | Article |
| id | doaj.art-a1fe0fb5ec98497fa9b242685d0c3736 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T16:38:25Z |
| publishDate | 2023-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-a1fe0fb5ec98497fa9b242685d0c37362023-11-24T14:54:28ZengMDPI AGMathematics2227-73902023-11-011122468310.3390/math11224683Metallic Structures for Tangent Bundles over Almost Quadratic <i>ϕ</i>-ManifoldsMohammad Nazrul Islam Khan0Sudhakar Kumar Chaubey1Nahid Fatima2Afifah Al Eid3Department of Computer Engineering, College of Computer, Qassim University, Buraydah 51452, Saudi ArabiaSection of Mathematics, Department of Information Technology, University of Technology and Applied Sciences, P.O. Box 77, Shinas 324, OmanDepartment of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi ArabiaDepartment of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi ArabiaThis paper aims to explore the metallic structure <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>J</mi><mn>2</mn></msup><mo>=</mo><mi>p</mi><mi>J</mi><mo>+</mo><mi>q</mi><mi>I</mi><mo>,</mo></mrow></semantics></math></inline-formula> where <i>p</i> and <i>q</i> are natural numbers, using complete and horizontal lifts on the tangent bundle <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mi>M</mi></mrow></semantics></math></inline-formula> over almost quadratic <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula>-structures (briefly, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>ϕ</mi><mo>,</mo><mi>ξ</mi><mo>,</mo><mi>η</mi><mo>)</mo></mrow></semantics></math></inline-formula>). Tensor fields <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>F</mi><mo>˜</mo></mover></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>F</mi><mo>*</mo></msup></semantics></math></inline-formula> are defined on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mi>M</mi></mrow></semantics></math></inline-formula>, and it is shown that they are metallic structures over <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>ϕ</mi><mo>,</mo><mi>ξ</mi><mo>,</mo><mi>η</mi><mo>)</mo></mrow></semantics></math></inline-formula>. Next, the fundamental 2-form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula> and its derivative <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>Ω</mo></mrow></semantics></math></inline-formula>, with the help of complete lift on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mi>M</mi></mrow></semantics></math></inline-formula> over <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>ϕ</mi><mo>,</mo><mi>ξ</mi><mo>,</mo><mi>η</mi><mo>)</mo></mrow></semantics></math></inline-formula>, are evaluated. Furthermore, the integrability conditions and expressions of the Lie derivative of metallic structures <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>F</mi><mo>˜</mo></mover></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>F</mi><mo>*</mo></msup></semantics></math></inline-formula> are determined using complete and horizontal lifts on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mi>M</mi></mrow></semantics></math></inline-formula> over <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>ϕ</mi><mo>,</mo><mi>ξ</mi><mo>,</mo><mi>η</mi><mo>)</mo></mrow></semantics></math></inline-formula>, respectively. Finally, we prove the existence of almost quadratic <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula>-structures on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mi>M</mi></mrow></semantics></math></inline-formula> with non-trivial examples.https://www.mdpi.com/2227-7390/11/22/4683metallic structuretangent bundlepartial differential equationsnijenhuis tensormathematical operatorslie derivatives |
| spellingShingle | Mohammad Nazrul Islam Khan Sudhakar Kumar Chaubey Nahid Fatima Afifah Al Eid Metallic Structures for Tangent Bundles over Almost Quadratic <i>ϕ</i>-Manifolds Mathematics metallic structure tangent bundle partial differential equations nijenhuis tensor mathematical operators lie derivatives |
| title | Metallic Structures for Tangent Bundles over Almost Quadratic <i>ϕ</i>-Manifolds |
| title_full | Metallic Structures for Tangent Bundles over Almost Quadratic <i>ϕ</i>-Manifolds |
| title_fullStr | Metallic Structures for Tangent Bundles over Almost Quadratic <i>ϕ</i>-Manifolds |
| title_full_unstemmed | Metallic Structures for Tangent Bundles over Almost Quadratic <i>ϕ</i>-Manifolds |
| title_short | Metallic Structures for Tangent Bundles over Almost Quadratic <i>ϕ</i>-Manifolds |
| title_sort | metallic structures for tangent bundles over almost quadratic i ϕ i manifolds |
| topic | metallic structure tangent bundle partial differential equations nijenhuis tensor mathematical operators lie derivatives |
| url | https://www.mdpi.com/2227-7390/11/22/4683 |
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