Almost Kenmotsu 3-manifolds satisfying some generalized nullity conditions. (English) Zbl 1488.53215
Summary: In this paper, a three-dimensional almost Kenmotsu manifold \(M^3\) satisfying the generalized \((\kappa,\mu)'\)-nullity condition is investigated. We mainly prove that on \(M^3\) the following statements are equivalent: (1) \(M^3\) is \(\phi\)-symmetric; (2) the Ricci tensor of \(M^3\) is cyclic-parallel; (3) the Ricci tensor of \(M^3\) is of Codazzi-type; (4) \(M^3\) is conformally flat with scalar curvature invariant along the Reeb vector field; (5) \(M^3\) is locally isometric to either the hyperbolic space \(\mathbb{H}^3(-1)\) or the Riemannian product \(\mathbb{H}^2(-4)\times \mathbb{R}\).
MSC:
| 53D15 | Almost contact and almost symplectic manifolds |
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
Keywords:
almost Kenmotsu 3-manifold; generalized \((\kappa,\mu)'\)-nullity condition; symmetry conditionReferences:
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