Generalized Quasi-Einstein Manifolds in Contact Geometry

In this study, we investigate generalized quasi-Einstein structure for normal metric contact pair manifolds. Firstly, we deal with elementary properties and examine, existence, and characterizations of generalized quasi-Einstein normal metric contact pair manifold. Secondly, the generalized quasi-constant curvature of normal metric contact pair manifolds are studied and it is proven that a normal metric contact pair manifold with generalized quasi-constant curvature is a generalized quasi-Einstein manifold. Normal metric contact pair manifolds satisfying cyclic Ricci tensor and the Codazzi type of Ricci tensor are considered and its shown that a generalized quasi-Einstein normal metric contact pair manifold does not satisfy Codazzi type of Ricci tensor. Finally, we work on normal metric contact pair manifolds satisfying certain curvature conditions related to $\mathcal{M}-$projective, conformal, and concircular curvature tensors. We show that a normal metric contact pair manifold with generalized quasi-constant curvature is locally isometric to the Hopf manifold $S^{2n+1}(1) \times S^1$.