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In the last paper, we introduced a new curvaturlike tensor field in an almost contact Riemannian manifold and we showed some geometrical properties of this tensor field in a Kenmotsu and a Sasakian manifold. In this paper, we define another new curvaturelike tensor field, named (CHR)3curvature tensor in an almost contact Riemannian manifold which is called a contact holomorphic Riemannian curvature tensor of the second type. Then, using this tensor, we mainly research (CHR)3-curvature tensor in a Sasakian manifold. Then we define the notion of the flatness of a (CHR)3-curvature tensor and we show that a Sasakian manifold with a flat (CHR)3-curvature tensor is flat. Next, we introduce the notion of (CHR)3-η-Einstein in an almost contact Riemannian manifold. In particular, we show that Sasakian (CHR)3η-Einstein manifold is η-Einstain. Moreover, we define the notion of (CHR)3space form and consider this in a Sasakian manifold. Finally, we consider a conformal transformation of an almost contact Riemannian manifold and we get new invariant tensor fields (not the conformal curvature tensor) under this transformation. Finally, we prove that a conformally (CHR)3-flat Sasakian manifold does not exist. 1. Almost contact Riemannian manifolds A real (2n + 1)-dimensional differentiable Riemannian manifold (M2n+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 φ2 = −I + η ⊗ ξ, η(φX) = 0, η(ξ) = 1, (1.1) g(φX, φY ) = g(X, Y ) − η(X)η(Y ) (1.2) for any Y, X ∈ TM2n+1, where ξ is defined by g(ξ, X) = η(X). From (1.1)3, the vector field ξ is unit and we say this vector field the structure vector field of 2010 Mathematics Subject Classification: 53C40.

A new curvaturelike tensor field in an almost contact Riemannian manifold II