Summary: In [J. Geom. 103, No. 1, 89–101 (2012; Zbl 1259.53029)] M. Prvanović considered several curvaturelike tensors defined for Hermitian manifolds. Developing her ideas in [the author, “A new curvaturelike tensor field in an almost contact Riemannian manifold II”, Publ. Inst. Math. (Beograd) (N.S.) 103, No. 117, 113–128 (2018; doi:10.2298/PIM1817113M)], we defined in an almost contact Riemannian manifold another new curvaturelike tensor field, which is called a contact holomorphic Riemannian curvature tensor or briefly \((CHR)_3\)-curvature tensor. Then, we mainly researched \((CHR)_3\)-curvature tensor in a Sasakian manifold. Also we proved, that a conformally \((CHR)_3\)-flat Sasakian manifold does not exist.
In the present paper, we consider this tensor field in a trans-Sasakian manifold. We calculate the \((CHR)_3\)-curvature tensor in a trans-Sasakian manifold. Also, the \((CHR)_3\)-Ricci tensor \(\rho_3\) and the \((CHR)_3\)-scalar curvature \(\tau_3\) in a trans-Sasakian manifold obtained.
Moreover, we define the notion of the \((CHR)_3\)-flatness in an almost contact Riemannian manifold. Then, we consider this notion in a trans-Sasakian manifold and determine the curvature tensor, the Ricci tensor and the scalar curvature. We prove that a \((CHR)_3\)-flat trans-Sasakian manifold is a generalized \(\eta\)-Einstein manifold. Finally, we obtain the expression of the curvature tensor with respect to the Riemannian metric \(g\) of a trans-Sasakian manifold, if the latter is \((CHR)_3\)-flat.