$*$-Weyl curvature tensor within the framework on Sasakian manifold admitting Zamkovoy connection

Document Type : Regular research papers

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Department of Mathematics, Bangalore University Bengaluru

Abstract

In this paper, we investigate $*$-Weyl curvature tensor of Sasakian manifold admitting Zamkovoy connection. If a sasakian manifold $M$ is Ricci flat with respect to the Zamkovoy connection then $M$ is an $\eta$-Einstein manifold. We study $*$-Weyl flat Sasakian manifold $M$ admitting Zamkovoy connection $\overline{\nabla}$ is an $\eta$-Einstein manifold. as well as $\xi$-$*$-Weyl flat Sasakian manifolds admitting Zamkovoy connection is an $\eta$-Einstein manifold, then its scalar curvature is constant. Additionally, we prove $\phi$-$*$-Weyl flat Sasakian manifolds admitting Zamkovoy connection then manifold is an $\eta$-Einstein manifold. Also $\xi$-$*$-Weyl flat with respect to Zamkovoy connection if and only if it is with respect Levi-Civita connection, provided that vector fields are horizantal vector fields. Moreover, We also prove that Sasakian manifolds satisfying $\overline{W}^{*}(\xi,U)\circ\overline{R}=0$, where $\overline{W}^{*}$ and $\overline{R}$ are $*$-Weyl curvature tensor and Riemannian curvature tensor with respect to Zamkovoy connection, meet specific conditions. Finally we conclude with an example for three-dimensional Sasakian manifolds.

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History

  • Receive Date: 10 January 2025
  • Revise Date: 17 April 2025
  • Accept Date: 30 June 2025

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