Geometric Analysis of η-Ricci Bourguignon Solitons on Para-Sasakian Manifolds With Semisymmetric Nonmetric Connection (SSNMC) on the Tangent Bundle

In this paper, we investigate the geometric properties of η-Ricci–Bourguignon (η-RB) solitons on para-Sasakian manifolds equipped with a semisymmetric nonmetric connection (SSNMC). By employing the complete lift on the tangent bundle, we derive curvature relations, Ricci identities, Ricci flow, and the corresponding η-RB soliton equations for the lifted manifold. It is shown that the complete lift of an η-RB soliton on a para-Sasakian manifold with SSNMC results in a generalized η-Einstein manifold. The nature of the soliton, such as shrinking, steady, or expanding, is characterized by the sign of the effective soliton coefficient α=λ+βr˜c−1/2. Furthermore, the conditions for Tθ-flat and Tθ–Ricci semisymmetric cases are established, illustrating how specific curvature constraints affect soliton behavior. An explicit example of a five-dimensional para-Sasakian manifold is constructed to verify the theoretical results using partial differential equations (PDEs). These findings extend the framework of η-RB solitons to manifolds with nonmetric connections and provide a foundation for future studies on geometric flows and higher-order lift structures.