Sasakian 3-Manifolds Satisfying Some Curvature Conditions Associated to Ƶ-Tensor
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In this paper, we study some curvature properties of Sasakian 3-manifolds associated to $\mathcal{Z}$-tensor. It is proved that if a Sasakian 3-manifold $(M,g)$ satisfies one of the conditions (1) the $\mathcal{Z}$-tensor is of Codazzi type, (2) $M$ is $\mathcal{Z}$-semisymmetric, (3) $M$ satisfies $Q(\mathcal{Z},R) = 0$, (4) $M$ is projectively $\mathcal{Z}$-semisymmetric, (5) $M$ is $\mathcal{Z}$-recurrent, then $(M,g)$ is of constant curvature 1. Several consequences are drawn from these results.
Abstract
In this paper, we study some curvature properties of Sasakian 3-manifolds associated to $\mathcal{Z}$-tensor. It is proved that if a Sasakian 3-manifold $(M,g)$ satisfies one of the conditions (1) the $\mathcal{Z}$-tensor is of Codazzi type, (2) $M$ is $\mathcal{Z}$-semisymmetric, (3) $M$ satisfies $Q(\mathcal{Z},R) = 0$, (4) $M$ is projectively $\mathcal{Z}$-semisymmetric, (5) $M$ is $\mathcal{Z}$-recurrent, then $(M,g)$ is of constant curvature 1. Several consequences are drawn from these results.
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