AbstractIn this paper, we study on normal complex contact metric manifold admitting a semi symmetric metric connection. We obtain curvature properties of a normal complex contact metric manifold admitting a semi symmetric metric connection. We also prove that this type of manifold is not conformal flat, concircular flat, and conharmonic flat. Finally, we examine complex Heisenberg group with the semi symmetric metric connection.

In this study, we investigate generalized quasi-Einstein normal metric contact pair manifolds. Initially, we deal with the elementary properties and existence of generalized quasi-Einstein normal metric contact pair manifolds. Later, we explore the generalized quasi-constant curvature of normal metric contact pair manifolds. It is proved 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 parallel Ricci tensor and the Codazzi type of Ricci tensor are considered, and further prove that a generalized quasi-Einstein normal metric contact pair manifold does not satisfy Codazzi type of Ricci tensor. Finally, we characterize normal metric contact pair manifolds satisfying certain curvature conditions related to 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 S2n+1(1)×S1.

Conformal, concircular, quasi-conformal and conharmonic curvature tensors play an important role in Riemannian geometry. In this paper, we study on normal complex contact metric manifolds under flatness conditions of these tensors.

In this paper, the geometry of normal metric contact pair manifolds is studied under the ‡atness of conformal, concircular and quasi-conformal curvature tensors. It is proved that a conformal ‡at normal metric contact pair manifold is an Einstein manifold with a positive scalar curvature and has positive sectional curvature. It is also shown that a concircular ‡at normal metric contact pair manifold is an Einstein manifold. Finally, it is obtained that a quasi-conformally ‡at normal metric contact pair manifold is an Einstein manifold with a positive scalar curvature and, is a space of constant curvature.

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