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Ricci tensor in 3-dimensional trans-Sasakian manifolds. (English) Zbl 1073.53060

J. A. Obunia studied a new class of almost contact metric structure called trans-Sasakian structure which is an analogue of locally conformal Kähler structure [Publ. Math. Debrecen, 32, 187–193 (1985)]. J. C. Marrero proved that a trans-Sasakian manifold of dimension \(\geq 5\) either cosymplecitc, or \(\alpha\)-Sasakian or \(\beta\)-Kenmotsu manifold and also constructed an example of 3-dimensional proper trans-Sasakian manifold [Ann. Mat. Pura Appl. 162, No. 4, 77–86 (1992; Zbl 0772.53036)].
D. Chinea and C. Gonzalez obtained some curvature identities for trans-Sasakian manifolds of dimension \(\geq 5\) [Proceedings of the XIIth Portuguese-Spanish Conference on Mathematics, Vol.II (Portugese, Braga), 569–571 (1987)].
In this paper, the authors obtain several explicit formulae for the Ricci operator, Ricci tensor and curvature tensor of a 3-dimensional trans-Sasakian manifold \((M,\phi,\xi,\eta,g)\) and applies them to the cases when the manifold being \(\eta\)-Einstein or satisfying \(R(X, Y)\cdot S= 0\), where \(R\) and \(S\) are the curvature tensor and the Ricci tensor of \(M\) respectively.

MSC:

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

Citations:

Zbl 0772.53036
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