Geometric and Structural Properties of Indefinite Kenmotsu Manifolds Admitting Eta-Ricci–Bourguignon Solitons

Geometric and Structural Properties of Indefinite Kenmotsu Manifolds Admitting Eta-Ricci–Bourguignon Solitons

This paper undertakes a detailed study of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-Ricci–Bourguignon solitons on <inline-formula&g...

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Bibliographic Details
Main Authors:	Md Aquib, Oğuzhan Bahadır, Laltluangkima Chawngthu, Rajesh Kumar
Format:	Article
Language:	English
Published:	MDPI AG 2025-06-01
Series:	Mathematics
Subjects:	<i>ϵ</i>-Kenmotsu manifold Ricci soliton <i>η</i>-Ricci–Bourguignon solitons <i>ϕ</i>-Ricci symmetric conharmonic curvature tensor
Online Access:	https://www.mdpi.com/2227-7390/13/12/1965
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Summary:	This paper undertakes a detailed study of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-Ricci–Bourguignon solitons on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϵ</mi></semantics></math></inline-formula>-Kenmotsu manifolds, with particular focus on three special types of Ricci tensors: Codazzi-type, cyclic parallel and cyclic <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-recurrent tensors that support such solitonic structures. We derive key curvature conditions satisfying Ricci semi-symmetric <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>R</mi><mo>·</mo><mi>E</mi><mo>=</mo><mn>0</mn><mo>)</mo></mrow></semantics></math></inline-formula>, conharmonically Ricci semi-symmetric <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>C</mi><mrow><mo>(</mo><mi>ξ</mi><mo>,</mo><msub><mi>β</mi><mi>X</mi></msub><mo>)</mo></mrow><mo>·</mo><mi>E</mi><mo>=</mo><mn>0</mn><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula>-projectively flat <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>P</mi><mrow><mo>(</mo><msub><mi>β</mi><mi>X</mi></msub><mo>,</mo><msub><mi>β</mi><mi>Y</mi></msub><mo>)</mo></mrow><mi>ξ</mi><mo>=</mo><mn>0</mn><mo>)</mo></mrow></semantics></math></inline-formula>, projectively Ricci semi-symmetric <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>L</mi><mo>·</mo><mi>P</mi><mo>=</mo><mn>0</mn><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>W</mi><mn>5</mn></msub></semantics></math></inline-formula>-Ricci semi-symmetric <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>W</mi><mrow><mo>(</mo><mi>ξ</mi><mo>,</mo><msub><mi>β</mi><mi>Y</mi></msub><mo>)</mo></mrow><mo>·</mo><mi>E</mi><mo>=</mo><mn>0</mn><mo>)</mo></mrow></semantics></math></inline-formula>, respectively, with the admittance of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-Ricci–Bourguignon solitons. This work further explores the role of torse-forming vector fields and provides a thorough characterization of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula>-Ricci symmetric indefinite Kenmotsu manifolds admitting <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-Ricci–Bourguignon solitons. Through in-depth analysis, we establish significant geometric constraints that govern the behavior of these manifolds. Finally, we construct explicit examples of indefinite Kenmotsu manifolds that satisfy the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-Ricci–Bourguignon solitons equation, thereby confirming their existence and highlighting their unique geometric properties. Moreover, these solitonic structures extend soliton theory to indefinite and physically meaningful settings, enhance the classification and structure of complex geometric manifolds by revealing how contact structures behave under advanced geometric flows and link the pure mathematical geometry to applied fields like general relativity. Furthermore, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-Ricci–Bourguignon solitons provide a unified framework that deepens our understanding of geometric evolution and structure-preserving transformations.
ISSN:	2227-7390