Hyperbolic ∗-Ricci Solitons and Gradient Hyperbolic ∗-Ricci Solitons on (ε)-Almost Contact Metric Manifolds of Type (α, β)

In this research paper, we introduce the notions of hyperbolic ∗-Ricci solitons and gradient hyperbolic ∗-Ricci solitons. We study the hyperbolic ∗-Ricci solitons on a three-dimensional ε-trans-Sasakian manifold. Specifically, we determine the hyperbolic ∗-Ricci solitons on a three-dimensional (ε)-trans-Sasakian manifold with a conformal vector field and a proper ϕ(Q*)-type vector field. Using hyperbolic ∗-Ricci solitons with a conformal vector field, we discuss some geometric symmetries on a three-dimensional (ε)-trans-Sasakian. In addition, we exhibit the nature of gradient hyperbolic ∗-Ricci solitons on a three-dimensional (ε)-trans-Sasakian manifold endowed with a scalar concircular field. Moreover, we demonstrate an example on a three-dimensional (ε)-trans-Sasakian manifold that admits the hyperbolic ∗-Ricci solitons and find the rate of change of the hyperbolic ∗-Ricci solitons within the same example. Lastly, we also introduce the concept of modified second hyperbolic ∗-Ricci solitons.