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.


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


Zbl 0772.53036