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

This paper undertakes a detailed study of η -Ricci–Bourguignon solitons on ϵ -Kenmotsu manifolds, with particular focus on three special types of Ricci tensors: Codazzi-type, cyclic parallel and cyclic η -recurrent tensors that support such solitonic structures. We derive key curvature conditions satisfying Ricci semi-symmetric ( R · E = 0 ) , conharmonically Ricci semi-symmetric ( C ( ξ , β X ) · E = 0 ) , ξ -projectively flat ( P ( β X , β Y ) ξ = 0 ) , projectively Ricci semi-symmetric ( L · P = 0 ) and W 5 -Ricci semi-symmetric ( W ( ξ , β Y ) · E = 0 ) , respectively, with the admittance of η -Ricci–Bourguignon solitons. This work further explores the role of torse-forming vector fields and provides a thorough characterization of ϕ -Ricci symmetric indefinite Kenmotsu manifolds admitting η -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 η -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, η -Ricci–Bourguignon solitons provide a unified framework that deepens our understanding of geometric evolution and structure-preserving transformations.