Abstract:
In this paper, we determine the components of the Weyl tensor of almost contact metric ($\mathrm{ACR-}$) manifold of class $C_{12}$ on associated $\mathrm{G}$-structure ($\mathrm{AG}$-structure) space. As an application, we prove that the conformally flat $\mathrm{ACR}$-manifold of class $C_{12}$ with $n>2$ is an $\eta$-Einstein manifold and conclude that it is an Einstein manifold such that the scalar curvature $r$ has provided. Also, the case when $n=2$ is discussed explicitly. Moreover, the relationships among conformally flat, conformally symmetric, $\xi$-conformally flat and $\Phi$-invariant Ricci tensor have been widely considered here and consequently we determine the value of scalar curvature $r$ explicitly with other applications. Finally, we define new classes with identities analogously to Gray identities and discuss their connections with class $C_{12}$ of $\mathrm{ACR}$-manifold.
Keywords:almost contact metric manifold of class $C_{12}$, $\eta$-Einstein manifold, Weyl tensor.
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