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In this paper, we investigate some curvature problems of Kenmotsu manifolds satisfying some certain conditions and we reach some classicifications. We consider -recurrent Kenmotsu manifolds and we show that -recurrent Kenmotsu manifolds are also -Einstein manifolds. Next, we study  -Ricci symmetric Kenmotsu manifolds and we find this manifolds are Einstein manifolds too. In addition, we examine locally -symmetric -Kenmotsu manifolds. Later we investigate this type manifold with quasi-conformally curvature tensor and concircular curvature tensor. In addition to these, we construct an example of Kenmotsu manifolds and we see that this example is a locally symmetric Kenmotsu manifold.   Key words: Kenmotsu manifold, -recurrent, -Ricci symmetric, locally -symmetric, concircular curvature tensor, quasi-conformally curvature tensor, -Einstein manifolds, Einstein manifolds.