Second Order Parallel Tensors on $\alpha $ - Sasakian Manifold

Abstract: Levy had proved that a second order symmetric parallel nonsingular tensor on a space of constant curvature is a constant multiple of the metric tensor. Sharma [12] has proved that a second order parallel tensor in a Kaehler space of constant holomorphic sectional curvature is a linear combination with constant coefficients of the Kaehlarian metric and the fundamental 2 - form. In this paper we show that a second order symmetric parallel tensor on an $\alpha $ - $K$ contact $\left( \alpha \in R_{o}\right) $ manifold is a constant multiple of the associated metric tensor and we also prove that there is no nonzero skew symmetric second order parallel tensor on an $\alpha $ - Sasakian manifold.

Keywords: Contact metric manifold, second order parallel tensor, K- contact and Sasakian manifold

Classification (MSC2000): 53C15; 53C25

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