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A New Curvaturelike Tensor Field in an Almost Contact Riemannian Manifold II
Koji Matsumoto
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 ${\left(\right)}_{3}$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 ${\left(\right)}_{3}$curvature tensor in a Sasakian manifold. Then we define the notion of the flatness of a ${\left(\right)}_{3}$curvature tensor and we show that a Sasakian manifold with a flat ${\left(\right)}_{3}$curvature tensor is flat. Next, we introduce the notion of ${\left(\right)}_{3}$$\eta $Einstein in an almost contact Riemannian manifold. In particular, we show that Sasakian ${\left(\right)}_{3}$$\eta $Einstein manifold is $\eta $Einstain. Moreover, we define the notion of ${\left(\right)}_{3}$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 ${\left(\right)}_{3}$flat Sasakian manifold does not exist.
Keywords: curvaturelike tensor field, almost contact Riemannian manifold, Sasakian manifold, ${\left(CHR\right)}_{3}$curvature tensor
Classification (MSC2000): 53C40
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Electronic fulltext finalized on: 26 Apr 2018.
This page was last modified: 11 Mai 2018.
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