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\(\eta\)-Einstein nearly Kenmotsu manifolds. (English) Zbl 1426.53052

Summary: In this paper, we show that an \(\eta\)-Einstein nearly Kenmotsu manifold with projective curvature tensor \(P\), and conharmonic curvature tensor \(N\), satisfy the conditions \(R(X, Y) \cdot P = 0\) and \(R(X, Y) \cdot N = 0\), respectively. Furthermore, we obtain scalar curvature of a projectively flat and a conharmonically flat \(\eta\)-Einstein nearly Kenmotsu manifold.

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53D15 Almost contact and almost symplectic manifolds
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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