Abstract. The paper examines two main topics: first, properties of the Yamabe soliton are explored by discussing a parallel vector field and demonstrating the validity of the Killing equation under curvature collineation. Second, the study of the gradient Yamabe soliton involves showing the skew-symmetry of the first two vector fields of the Hessian curvature tensor and deriving the soliton equation's expression in affine collineation.

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