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. 
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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/

[1] Abood H. M., Abass M. Y., “A study of new class of almost contact metric manifolds of Kenmotsu type”, Tamkang Journal of Mathematics, 52:2 (2021), 253–266 | DOI | MR | Zbl

[2] Abass M. Y., Geometry of certain curvature tensors of almost contact metric manifold, PhD thesis, College of Education for Pure Sciences, University of Basrah, 2020 https://faculty.uobasrah.edu.iq/uploads/publications/1620305696.pdf

[3] Abu-Saleem A., Rustanov A. R., Kharitonova S. V., “Axiom of $\Phi$-holomorphic ($2r+1$)-planes for generalized Kenmotsu manifolds”, Vestnik Tomskogo Gosudarstvennogo Universiteta. Matematika i Mekhanika, 2020, no. 66, 5–23 (in Russian) | DOI

[4] Alegre P., Fernández L. M., Prieto-Martín A., “A new class of metric $f$-manifolds”, Carpathian Journal of Mathematics, 34:2 (2018), 123–134 https://www.jstor.org/stable/26898721 | DOI | MR | Zbl

[5] Blair D. E., Yıldırım H., “On conformally flat almost contact metric manifolds”, Mediterranean Journal of Mathematics, 13:5 (2016), 2759–2770 | DOI | MR | Zbl

[6] Bouzir H., Beldjilali G., Bayour B., “On three dimensional $C_{12}$-manifolds”, Mediterranean Journal of Mathematics, 18:6 (2021), 239 | DOI | MR | Zbl

[7] de Candia S., Falcitelli M., “Even-dimensional slant submanifolds of a $C_{5}\oplus C_{12}$-manifold”, Mediterranean Journal of Mathematics, 14:6 (2017), 224 | DOI | MR | Zbl

[8] de Candia S., Falcitelli M., “Curvature of $C_{5}\oplus C_{12}$-manifold”, Mediterranean Journal of Mathematics, 16:4 (2019), 105 | DOI | MR | Zbl

[9] Chinea D., Gonzalez C., “A classification of almost contact metric manifolds”, Annali di Matematica Pura ed Applicata, 156:1 (1990), 15–36 | DOI | MR | Zbl

[10] Chojnacka-Dulas J., Deszcz R., Głogowska M., Prvanović M., “On warped product manifolds satisfying some curvature conditions”, Journal of Geometry and Physics, 74 (2013), 328–341 | DOI | MR | Zbl

[11] De U. C., Suh Y. J., “On weakly semiconformally symmetric manifolds”, Acta Mathematica Hungarica, 157:2 (2019), 503–521 | DOI | MR | Zbl

[12] Deszcz R., Głogowska M., Jełowicki J., Zafindratafa G., “Curvature properties of some class of warped product manifolds”, International Journal of Geometric Methods in Modern Physics, 13:01 (2016), 1550135 | DOI | MR | Zbl

[13] Hwang S., Yun G., “Ridigity of Ricci solitons with weakly harmonic Weyl tensors”, Mathematische Nachrichten, 291:5–6 (2018), 897–907 | DOI | MR | Zbl

[14] Kenmotsu K., “A class of almost contact Riemannian manifolds”, T{ō}hoku Mathematical Journal, 24:1 (1972), 93–103 | DOI | MR | Zbl

[15] Kirichenko V. F., Differential-geometric structures on manifolds, Pechatnyi Dom, Odessa, 2013

[16] Kirichenko V. F., Dondukova N. N., “Contactly geodesic transformations of almost-contact metric structures”, Mathematical Notes, 80:1–2 (2006), 204–213 | DOI | DOI | Zbl

[17] Kirichenko V. F., Kusova E. V., “On geometry of weakly cosymplectic manifolds”, Journal of Mathematical Sciences, 177:5 (2011), 668–674 | DOI | MR | Zbl

[18] Rustanov A. R., Polkina E. A., Kharitonova S. V., “Projective invariants of almost $C(\lambda)$-manifolds”, Annals of Global Analysis and Geometry, 61:2 (2022), 459–467 | DOI | MR | Zbl

[19] Venkatesha, Naik D. M., Kumara H. A., “Conformal curvature tensor on paracontact metric manifolds”, Matematic̆ki Vesnik, 72:3 (2020), 215–225 http://www.vesnik.math.rs/vol/mv20304.pdf | MR | Zbl

[20] Wang Y., Wang W., “Curvature properties of almost Kenmotsu manifolds with generalized nullity conditions”, Filomat, 30:14 (2016), 3807–3816 | DOI | MR | Zbl