Geodesic
On Weyl tensor of $\mathrm{ACR}$-manifolds of class $C_{12}$ with applications
Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 59 (2022), pp. 3-14

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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. 
@article{IIMI_2022_59_a0,
     author = {M. Y. Abass and Q. S. Al-Zamil},
     title = {On {Weyl} tensor of $\mathrm{ACR}$-manifolds of class $C_{12}$ with applications},
     journal = {Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta},
     pages = {3--14},
     publisher = {mathdoc},
     volume = {59},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IIMI_2022_59_a0/}
}
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M. Y. Abass; Q. S. Al-Zamil. On Weyl tensor of $\mathrm{ACR}$-manifolds of class $C_{12}$ with applications. Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 59 (2022), pp. 3-14. http://geodesic.mathdoc.fr/item/IIMI_2022_59_a0/