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On \(\eta\)-Einstein trans-Sasakian manifolds. (English) Zbl 1249.53037

Summary: A systematic study of \(\eta\)-Einstein trans-Sasakian manifold is performed. We find eight necessary and sufficient conditions for the structure vector field \(\xi\) of a trans-Sasakian manifold to be an eigenvector field of the Ricci operator. We show that for a 3-dimensional almost contact metric manifold \((M,\varphi,\xi,\eta,g)\), the conditions of being normal, trans-\(K\)-contact, trans-Sasakian are all equivalent to \(\nabla\xi\circ\varphi =\varphi\circ\nabla\xi\). In particular, the conditions of being quasi-Sasakian, normal with \(0=2\beta =\div\xi\), trans-\(K\)-contact of type \((\alpha,0)\), trans-Sasakian of type \((\alpha,0)\), and \(\mathcal C_6\)-class are all equivalent to \(\nabla\xi=-\alpha\varphi\), where \(2\alpha={\text{Trace}}(\varphi\nabla\xi)\). In last, we give fifteen necessary and sufficient conditions for a 3-dimensional trans-Sasakian manifold to be \(\eta\)-Einstein.

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C35 Differential geometry of symmetric spaces
53D10 Contact manifolds (general theory)
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