A (CHR)3-flat trans-Sasakian manifold

In [4] M. Prvanovic considered several curvaturelike tensors 
defined for Hermitian manifolds. Developing her ideas in [3], 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 ρ3  and the (CHR)3-scalar curvature τ3  in a trans-Sasakian manifold have been 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 proved that a (CHR)3-flat trans-Sasakian manifold is a generalized   ɳ-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.


ALMOST CONTACT RIEMANNIAN MANIFOLDS
A real (2n+1)-dimensional differentiable Riemannian manifold (M 2n+1 , g) is said to be an almost contact Riemannian manifold if it has a (1, 1)-tensor φ and a 1-form η which satisfy for any Y, X P T M 2n+1 , where ξ is defined by g(ξ, X) = η(X) and T M 2n+1 is the tangent bundle of M 2n+1 . From (1.1) 3 , the vector field ξ is unit and we call this vector field the structure vector field of the almost contact Riemannian manifold. Next, in an almost contact Riemannian manifold M 2n+1 we define a 2-form F by for all X, Y P T M 2n+1 . Then the 2-form F is skew-symmetric and we call this tensor field the fundamental 2-form of this almost contact Riemannian manifold. Hereafter, we write the same φ instead of F . An almost contact manifold M 2n+1 is called trans-Sasakian if the fundamental form φ satisfies (∇ X φ)(Y, Z) = αtg(X, Y )η(Z)´g(X, Z)η(Y )u+ + βtφ(X, Y )η(Z)´φ(X, Z)η(Y )u, (1.4) for certain smooth functions α and β on M 2n+1 and for all tangetn vectors X, Y, Z P T M 2n+1 , where ∇ means the covariant differentiation with respect to g. In that case we will say that a trans-Sasakian structure is of type (α, β) or of an (α, β)-type, [5].
On trans-Sasakian manifolds 13 In a trans-Sasakian manifold of (α, β)-type, we know, [5], that for any X, Y P T M 2n+1 , where ρ is the Ricci tensor with respect to g and A(X, Y, Z) is defined as for any X, Y, Z P T M 2n+1 . The following equations (1.7), (1.8), (1.9), (1.10) and (1.11) are very useful for calculations of the (CHR) 3 -curvature tensor in a trans-Sasakian manifold.

(CHR) 3 -CURVATURE TENSOR IN A TRANS-SASAKIAN MANIFOLD
In this section, we consider the (CHR) 3 -curvature tensor in a trans-Sasakian manifold.
Since, (B) is the equation which change X ô Y and Z ô W in (A), we have ] .
By virtue of the above two equations, we obtain ] .
Let us consider a (CHR) 3 -flat trans-Sasakian manifold. Then the left hand side of (2.10) is zero.
Next, since we have we obtain ) .