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Almost Kenmotsu 3-manifolds satisfying some generalized nullity conditions

Wang Wenjie (School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning, P. R. China)
Liu Ximin (School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning, P. R. China)

In this paper, a three-dimensional almost Kenmotsu manifold M3 satisfying the generalized (k,μ)'-nullity condition is investigated. We mainly prove that on M3 the following statements are equivalent: (1) M3 is Φ-symmetric; (2) the Ricci tensor of M3 is cyclic-parallel; (3) the Ricci tensor of M3 is of Codazzi type; (4) M3 is conformally flat with scalar curvature invariant along the Reeb vector field; (5) M3 is locally isometric to either the hyperbolic space H3(-1) or the Riemannian product H2(-4) x R.

Keywords: Almost Kenmotsu 3-manifold, generalized (k,μ)'-nullity condition, symmetry condition