Real Hypersurfaces in ℂP 2 and ℂH 2 with Cyclic Parallel ∗-Ricci Tensor

In this paper, we prove that the ∗-Ricci tensor of a real hypersurface in complex projective plane ℂP 2 or complex hyperbolic plane ℂH 2 is cyclic parallel if and only if the hypersurface is of type (A). We find some three-dimensional real hypersurfaces having non-vanishing and non-parallel ∗-Ricci tensors which are cyclic parallel.

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.

On static manifolds and related critical spaces with cyclic parallel Ricci tensor

We classify 3-dimensional compact Riemannian manifolds (M 3, g) that admit a non-constant solution to the equation −Δfg +Hess f − f Ric = μ Ric +λg for some special constants (μ, λ), under the assumption that the manifold has cyclic parallel Ricci tensor. Namely, the structures that we study here are: positive static triples, critical metrics of the volume functional, and critical metrics of the total scalar curvature functional. We also classify n-dimensional critical metrics of the volume functional with non-positive scalar curvature and satisfying the cyclic parallel Ricci tensor condition.

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.

Cyclic-parallel Ricci tensors on a class of almost Kenmotsu 3-manifolds

In this paper, we give a local characterization for the Ricci tensor of an almost Kenmotsu [Formula: see text]-manifold [Formula: see text] to be cyclic-parallel. As an application, we prove that if [Formula: see text] has cyclic-parallel Ricci tensor and satisfies [Formula: see text], (where [Formula: see text] is the Lie derivative of [Formula: see text] along the Reeb flow and both [Formula: see text] and [Formula: see text] are smooth functions such that [Formula: see text] is invariant along the contact distribution), then [Formula: see text] is locally isometric to either the hyperbolic space [Formula: see text] or a non-unimodular Lie group equipped with a left invariant non-Kenmotsu almost Kenmotsu structure.

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.

η-Ricci Solitons on Sasakian 3-Manifolds

In this paper we studyη-Ricci solitons on Sasakian 3-manifolds. Among others we prove that anη-Ricci soliton on a Sasakian 3-manifold is anη-Einstien manifold. Moreover we considerη-Ricci solitons on Sasakian 3-manifolds with Ricci tensor of Codazzi type and cyclic parallel Ricci tensor. Beside these we study conformally flat andφ-Ricci symmetricη-Ricci soliton on Sasakian 3-manifolds. Alsoη-Ricci soliton on Sasakian 3-manifolds with the curvature conditionQ.R= 0 have been considered. Finally, we construct an example to prove the non-existence of properη-Ricci solitons on Sasakian 3-manifolds and verify some results.

On a type of spacetime

The object of the present paper is to study a spacetime admitting conharmonic curvature tensor and some geometric properties related to this spacetime. It is shown that in a conharmonically flat spacetime with cyclic parallel Ricci tensor, the energy–momentum tensor is cyclic parallel and conversely. Finally, we prove that for a radiative perfect fluid spacetime if the energy–momentum tensor satisfying the Einstein’s equations without cosmological constant is generalized recurrent, then the fluid has vanishing vorticity and the integral curves of the vector field [Formula: see text] are geodesics.