A New Curvaturelike Tensor Field in an Almost Contact Riemannian Manifold II

Koji Matsumoto

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

Full text of the article: (for faster download, first choose a mirror)

• Compressed DVI file (28 kilobytes)
• Compressed PostScript file (380 kilobytes)
• PDF file (400 kilobytes)