Almost Ricci Solitons on Co-K¨ ahler Manifolds

Abstract

This paper investigates the geometrical structure of co-K¨ ahler man ifolds admitting semi-symmetric metric ξ-connection (SSM ξ-connection). We derive several fundamental identities involving the curvature tensor, scalar cur vature, and Ricci operator that characterize the geometry of such manifolds. In particular, we identify the criteria for the manifold to be η-Einstein and exam ine the behavior of Ricci soliton structures in such framework. We prove that when a co-K¨ ahler manifold with SSM ξ-connection admit a Ricci soliton, then the soliton is necessarily expanding. Furthermore, we reveal that an η-Einstein co-K¨ ahler manifold with such a connection admitting Ricci soliton structure is Einstein. Additionally, we investigate three-dimensional co-K¨ahler manifolds admitting Ricci soliton with a SSM ξ-connection, and conclude with an ex amination of co-K¨ ahler manifolds equipped with such a connection admitting gradient Ricci almost solitons.