A new curvature-like tensor in an almost contact Riemannian manifold

In a M. Prvanović's paper [5], we can find a new curvature-like tensor in an almost Hermitian manifold. In this paper, we define a new curvature-like tensor, named contact holomorphic Riemannian, briefly (CHR), curvature tensor in an almost contact Riemannian manifold. Then, using this tensor, we mainly research (CHR)- curvature tensor in a Kenmotsu and a Sasakian manifold. We introduce the flatness of a (CHR)-curvature tensor and show that a Kenmotsu and a Sasakian manifold with a flat (CHR)-curvature tensor is flat, see Theorems 3.1 and 4.1. Next, we introduce the notion of an (CHR)--Einstein in an almost contact Riemannian manifold. In particular, in a Sasakian or a Kenmotsu manifold, a (CHR)--Einstein manifold is -Einstein, see Theorem 5.3. Finally, from this tensor, we introduce a notion of a (CHR)-space form in an almost contact Riemannian manifold. In particular, if a Kenmotsu and a Sasakian manifold are (CHR)-space form, then the (CHR)-curvature tensor satisfies a special equation, see Theorems 6.2 and 7.1.