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On three-dimensional trans-Sasakian manifolds. (English) Zbl 1175.53058

Let \(M\) be a three-dimensional (connected) trans-Sasakian manifold of type \((\alpha,\beta)\) with \(\alpha\), \(\beta\) constants. In this paper the authors prove that the following properties are equivalent:
(1) \(M\) is locally \(\phi\)-symmetric;
(2) \(M\) has \(\eta\)-parallel Ricci tensor;
(3) \(M\) has constant scalar curvature.
An example of a three-dimensional locally \(\phi\)-symmetric trans-Sasakian of type \((0, 1)\) is given.
Reviewer: D. Perrone (Lecce)

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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