Some curvature tensors on a trans-Sasakian manifold. (English) Zbl 1138.53028

A trans-Sasakian structure on a manifold \(M\) is an almost contact metric structure whose cone \(M\times\mathbb R\) belongs to the class \(W_4\) in the classification of Gray and Hervella. In this paper, this kind of structure is studied under the assumptions of projective semisymmetry, Weyl semisymmetry and concircular semisymmetry. By studying the action of the curvature tensor \(R(X,Y)\), considered as a derivation of tensor algebra, one gets examples of Einstein manifolds, \(\eta\)-Einstein, conformally flat and concircularly flat manifolds.


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