On trans-Sasakian $3$-manifolds as $\eta$-Einstein solitons

The present paper is to deliberate the class of $3$-dimensional trans-Sasakian manifolds which admits $\eta$-Einstein solitons. We have studied $\eta$-Einstein solitons on $3$-dimensional trans-Sasakian manifolds where the Ricci tensors are Codazzi type and cyclic parallel. We have also discussed some curvature conditions admitting $\eta$-Einstein solitons on $3$-dimensional trans-Sasakian manifolds and the vector field is torse-forming. We have also shown an example of $3$-dimensional trans-Sasakian manifold with respect to $\eta$-Einstein soliton to verify our results.

Real hypersurfaces in the complex hyperbolic quadric with normal Jacobi operator of Codazzi type

In this paper, we introduce the notion of normal Jacobi operator of Codazzi type for real hypersurfaces in the complex hyperbolic quadric [Formula: see text]. The normal Jacobi operator of Codazzi type implies that the unit normal vector field [Formula: see text] becomes [Formula: see text]-principal or [Formula: see text]-isotropic. Then according to each case, we give a complete classification of Hopf real hypersurfaces in [Formula: see text] with normal Jacobi operator of Codazzi type. The result of the classification shows that no such hypersurfaces exist.

ETA-RICCI SOLITONS ON LORENTZIAN PARA-KENMOTSU MANIFOLDS

The objective of present research article is to investigate the geometric properties of $\eta$-Ricci solitons on Lorentzian para-Kenmotsu manifolds. In this manner, we consider $\eta$-Ricci solitons on Lorentzian para-Kenmotsu manifolds satisfying $R\cdot S=0$. Further, we obtain results for $\eta$-Ricci solitons on Lorentzian para-Kenmotsu manifolds with quasi-conformally flat property. Moreover, we get results for $\eta$-Ricci solitons in Lorentzian para-Kenmotsu manifolds admitting Codazzi type of Ricci tensor and cyclic parallel Ricci tensor, $\eta$-quasi-conformally semi-symmetric, $\eta$-Ricci symmetric and quasi-conformally Ricci semi-symmetric. At last, we construct an example of a such manifold which justify the existence of proper $\eta$-Ricci solitons.

Curvature Properties of Generalized pp-Wave Metrics

The main objective of the present paper is to investigate the curvature properties of generalized pp-wave metrics. It is shown that a generalized pp-wave spacetime is Ricci generalized pseudosymmetric, 2-quasi-Einstein and generalized quasi-Einstein in the sense of Chaki. As a special case it is shown that pp-wave spacetime is semisymmetric, semisymmetric due to conformal and projective curvature tensors, R-space by Venzi and satisfies the pseudosymmetric type condition P ⋅ P = −13Q(S,P). Again we investigate the sufficient condition for which a generalized pp-wave spacetime turns into pp-wave spacetime, pure radiation spacetime, locally symmetric and recurrent. Finally, it is shown that the energy-momentum tensor of pp-wave spacetime is parallel if and only if it is cyclic parallel. Again the energy momentum tensor is Codazzi type if it is cyclic parallel but the converse is not true as shown by an example. Finally, we make a comparison between the curvature properties of the Robinson-Trautman metric and generalized pp-wave metric.

Generalized Quasi-Einstein Manifolds in Contact Geometry

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.

There are no genuine Kähler–Codazzi manifolds

Nearly Kähler- and Kähler–Codazzi-type manifolds are defined in a very similar way. We prove that nearly Kähler-type manifolds make sense only in Hermitian and para-Hermitian contexts, and that Kähler–Codazzi-type manifolds reduce to Kähler-type manifolds in all the four Hermitian, para-Hermitian, Norden and product Riemannian geometries. Kähler–Codazzi condition is also studied on almost complex golden manifolds.

On a type of spacetimes

In the present paper we characterize a type of spacetimes, called almost pseudo Z-symmetric spacetimes A(PZS)4. At first, we obtain a condition for an A(PZS)4 spacetime to be a perfect fluid spacetime and Roberson-Walker spacetime. It is shown that an A(PZS)4 spacetime is a perfect fluid spacetime if the Z tensor is of Codazzi type. Next we prove that such a spacetime is the Roberson-Walker spacetime and can be identified with Petrov types I, D or O[3], provided the associated scalar ? is constant. Then we investigate A(PZS)4 spacetimes satisfying divC = 0 and state equation is derived. Also some physical consequences are outlined. Finally, we construct a metric example of an A(PZS)4 spacetime.

SOME RESULTS ON GENERALIZED $(k,\mu)$-PARACONTACT METRIC MANIFOLDS

The object of this paper is to study Codazzi type of Ricci tensor in generalized $(k,\mu )$-paracontact metric manifolds. Next we study cyclic parallel Ricci tensor in generalized $(k,\mu )$-paracontact metric manifolds. Further, we characterized generalized $(k,\mu )$-paracontact metric manifolds whose structure tensor $\phi$ is $\eta$-parallel. Finally, we investigate locally $\phi$-Ricci symmetric generalized $(k,\mu )$-paracontact metric manifolds.

η-Ricci Solitons on Kenmotsu 3-Manifolds

In the present paper we study η-Ricci solitons on Kenmotsu 3-manifolds. Moreover, we consider η-Ricci solitons on Kenmotsu 3-manifolds with Codazzi type of Ricci tensor and cyclic parallel Ricci tensor. Beside these, we study φ-Ricci symmetric η-Ricci soliton on Kenmotsu 3-manifolds. Also Kenmotsu 3-manifolds satisfying the curvature condition R.R = Q(S, R)is considered. Finally, an example is constructed to prove the existence of a proper η-Ricci soliton on a Kenmotsu 3-manifold.