<i>η</i>-Ricci–Yamabe Solitons along Riemannian Submersions
In this paper, we investigate the geometrical axioms of Riemannian submersions in the context of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-for...
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MDPI AG
2023-08-01
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| author | Mohd Danish Siddiqi Fatemah Mofarreh Mehmet Akif Akyol Ali H. Hakami |
| author_facet | Mohd Danish Siddiqi Fatemah Mofarreh Mehmet Akif Akyol Ali H. Hakami |
| author_sort | Mohd Danish Siddiqi |
| collection | DOAJ |
| description | In this paper, we investigate the geometrical axioms of Riemannian submersions in the context of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-Ricci–Yamabe soliton (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-RY soliton) with a potential field. We give the categorization of each fiber of Riemannian submersion as an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-RY soliton, an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-Ricci soliton, and an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-Yamabe soliton. Additionally, we consider the many circumstances under which a target manifold of Riemannian submersion is an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-RY soliton, an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-Ricci soliton, an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-Yamabe soliton, or a quasi-Yamabe soliton. We deduce a Poisson equation on a Riemannian submersion in a specific scenario if the potential vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula> of the soliton is of gradient type =:grad<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>γ</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> and provide some examples of an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-RY soliton, which illustrates our finding. Finally, we explore a number theoretic approach to Riemannian submersion with totally geodesic fibers. |
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| spelling | doaj.art-6597fd525ab049fdacb788f5b6029e4b2023-11-19T00:15:24ZengMDPI AGAxioms2075-16802023-08-0112879610.3390/axioms12080796<i>η</i>-Ricci–Yamabe Solitons along Riemannian SubmersionsMohd Danish Siddiqi0Fatemah Mofarreh1Mehmet Akif Akyol2Ali H. Hakami3Department of Mathematics, College of Science, Jazan University, P.O. Box 277, Jazan 45142, Saudi ArabiaMathematical Science Department, Faculty of Science, Princess Nourah bint Abdulrahman University, Riyadh 11546, Saudi ArabiaDepartment of Mathematics, Faculty of Arts and Sciences, Bingol University, Bingol 12000, TurkeyDepartment of Mathematics, College of Science, Jazan University, P.O. Box 277, Jazan 45142, Saudi ArabiaIn this paper, we investigate the geometrical axioms of Riemannian submersions in the context of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-Ricci–Yamabe soliton (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-RY soliton) with a potential field. We give the categorization of each fiber of Riemannian submersion as an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-RY soliton, an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-Ricci soliton, and an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-Yamabe soliton. Additionally, we consider the many circumstances under which a target manifold of Riemannian submersion is an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-RY soliton, an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-Ricci soliton, an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-Yamabe soliton, or a quasi-Yamabe soliton. We deduce a Poisson equation on a Riemannian submersion in a specific scenario if the potential vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula> of the soliton is of gradient type =:grad<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>γ</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> and provide some examples of an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-RY soliton, which illustrates our finding. Finally, we explore a number theoretic approach to Riemannian submersion with totally geodesic fibers.https://www.mdpi.com/2075-1680/12/8/796<i>η</i>-Ricci–Yamabe solitonRiemannian submersionRiemannian manifoldhomotopy groups |
| spellingShingle | Mohd Danish Siddiqi Fatemah Mofarreh Mehmet Akif Akyol Ali H. Hakami <i>η</i>-Ricci–Yamabe Solitons along Riemannian Submersions Axioms <i>η</i>-Ricci–Yamabe soliton Riemannian submersion Riemannian manifold homotopy groups |
| title | <i>η</i>-Ricci–Yamabe Solitons along Riemannian Submersions |
| title_full | <i>η</i>-Ricci–Yamabe Solitons along Riemannian Submersions |
| title_fullStr | <i>η</i>-Ricci–Yamabe Solitons along Riemannian Submersions |
| title_full_unstemmed | <i>η</i>-Ricci–Yamabe Solitons along Riemannian Submersions |
| title_short | <i>η</i>-Ricci–Yamabe Solitons along Riemannian Submersions |
| title_sort | i η i ricci yamabe solitons along riemannian submersions |
| topic | <i>η</i>-Ricci–Yamabe soliton Riemannian submersion Riemannian manifold homotopy groups |
| url | https://www.mdpi.com/2075-1680/12/8/796 |
| work_keys_str_mv | AT mohddanishsiddiqi iēiricciyamabesolitonsalongriemanniansubmersions AT fatemahmofarreh iēiricciyamabesolitonsalongriemanniansubmersions AT mehmetakifakyol iēiricciyamabesolitonsalongriemanniansubmersions AT alihhakami iēiricciyamabesolitonsalongriemanniansubmersions |