We consider η-Ricci solitons on Lorentzian para-Sasakian manifolds satisfying certain curvature conditions: R(ξ,X) · S = 0 and S · R(ξ,X) = 0. We prove that on a Lorentzian para-Sasakian manifold (M, φ, ξ, η, 1), if the Ricci curvature satisfies one of the previous conditions, the existence of η-Ricci solitons implies that (M, 1) is Einstein manifold. We also conclude that in these cases there is no Ricci soliton on M with the potential vector field ξ. On the other way, if M is of constant curvature, then (M, 1) is elliptic manifold. Cases when the Ricci tensor satisfies different other conditions are also discussed.

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η-Ricci solitons on Lorentzian para-Sasakian manifolds

The object of the present paper is to introduce a non-flat Riemannian manifold called hyper-generalized recurrent manifolds and study its various geometric properties along with the existence of a proper example.

On a generalized class of recurrent manifolds

A second-order differential identity for the Riemann tensor is obtained, on a manifold with a symmetric connection. Several old and some new differential identities for the Riemann and Ricci tensors are derived from it. Applications to manifolds with recurrent or symmetric structures are discussed. The new structure of K-recurrency naturally emerges from an invariance property of an old identity due to Lovelock.

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A second-order identity for the Riemann tensor and applications

Let the metric $g$ of a three-dimensional $\eta$-Einstein almost Kenmotsu manifold $M$ be a Ricci soliton, we prove that $M$ is a Kenmotsu manifold of constant sectional curvature $-1$ and the soliton is expanding.

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RICCI SOLITONS ON THREE-DIMENSIONAL $\eta$-EINSTEIN ALMOST KENMOTSU MANIFOLDS

A second-order differential identity for the Riemann tensor is obtained, on a manifold with symmetric connection. Several old and some new differential identities for the Riemann and Ricci tensors descend from it. Applications to manifolds with Recurrent or Symmetric structures are discussed. The new structure of K-recurrency naturally emerges from an invariance property of an old identity by Lovelock.

A second-order identity for the Riemann tensor

The aim of the present paper is to study 3-dimensional f-Kenmotsu manifolds and Ricci solitons. First, we give an example of a 3-dimensional f-Kenmotsu manifold. Then we consider a Riccisemisymmetric 3-dimensional f-Kenmotsu manifold and prove that a 3-dimensional f-Kenmotsu manifold is Ricci semisymmetric if and only if it is an Einstein manifold. Moreover, we investigate an η-parallel Ricci tensor in a 3-dimensional f-Kenmotsu manifold. Finally, we study Ricci solitons in a 3-dimensional f-Kenmotsu manifold.

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On 3-dimensional f-Kenmotsu manifolds and Ricci solitons

The object of the present paper is to study a type of Riemannian manifolds called generalized recurrent manifolds. We have constructed two concrete examples of such a manifold whose scalar curvature is non-zero non-constant. Some other properties have been considered. Among others it is shown that on a generalized recurrent manifold with constant scalar curvature, Weyl-semisymmetry and semisymmetry are equivalent. Sufficient condition for a generalized recurrent manifold to be a special quasi Einstein manifold is obtained.

On generalized recurrent Riemannian manifolds

The object of the present paper is to study 3-dimensional trans-Sasakian manifolds admitting Ricci solitons and gradient Ricci solitons. We prove that if (1,V, λ) is a Ricci soliton where V is collinear with the characteristic vector field ξ, then V is a constant multiple of ξ and the manifold is of constant scalar curvature provided α, β =constant. Next we prove that in a 3-dimensional trans-Sasakian manifold with constant scalar curvature if 1 is a gradient Ricci soliton, then the manifold is either a β-Kenmotsu manifold or an Einstein manifold. As a consequence of this result we obtain several corollaries.

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Ricci solitons and gradient Ricci solitons in three-dimensional trans-Sasakian manifolds

The object of this paper is to study φ -recurrent Kenmotsu manifolds. Also three-dimensional locally φ - recurrent Kenmotsu manifolds have been considered. Among others it is proved that a locally φ -recurrent Kenmotsu spacetime is the Robertson-Walker spacetime. Finally we give a concrete example of a three- dimensional Kenmotsu manifold.

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Ahmet Yildiz

Papers

On 3-dimensional f-Kenmotsu manifolds and Ricci solitons

On generalized recurrent Riemannian manifolds

Ricci solitons and gradient Ricci solitons in three-dimensional trans-Sasakian manifolds

On f-Recurrent Kenmotsu Manifolds

Locally ϕ-symmetric normal almost contact metric manifolds of dimension 3