Exploring Gradient Soliton Structures and Metric Deformations on Riemannian Manifolds with Parallel Vector Fields

Abstract

In this present paper, we delve into an investigation of the distinctive characteristics and properties exhibited by various Ricci soliton structures that manifest within Riemannian manifolds equipped with parallel vector fields. Expanding upon these discoveries, we offer systematic classifications related to the conformal and semi-conformal modifications applied to a Riemannian metric denoted as $g$. To enhance the clarity and practical applicability of our research, we provide illustrative examples that serve to validate and exemplify the theoretical findings and relationships presented throughout the paper. These examples serve as tangible instances that concretely exemplify the presence and behavior of manifolds endowed with the aforementioned soliton structures.