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A New Curvaturelike Tensor Field in an Almost Contact Riemannian Manifold II
Department of Mathematics, Yamagata University, Yamagata, Japan
Abstract: 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 -curvature 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 -curvature tensor in a Sasakian manifold. Then we define the notion of the flatness of a -curvature tensor and we show that a Sasakian manifold with a flat -curvature tensor is flat. Next, we introduce the notion of --Einstein in an almost contact Riemannian manifold. In particular, we show that Sasakian --Einstein manifold is -Einstain. Moreover, we define the notion of -space 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 -flat Sasakian manifold does not exist.
Keywords: curvaturelike tensor field, almost contact Riemannian manifold, Sasakian manifold, -curvature tensor
Classification (MSC2000): 53C40
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