Geodesic
Curvature identities for principle $T^1$-bundles over almost Hermitian manifolds
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2010), pp. 56-63.

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We study the equivalence of identities $R_1$, $R_2$, and $R_3$ for an almost Hermitian structure $S$ on the base of a canonical principal $T^1$-bundle and their contact analogs for the induced almost contact metric structure $S^\sharp$ on the total space of this bundle. We prove that the canonical connection of a canonical principal $T^1$-bundle over a Hermitian or a quasi-Kählerian manifold of the class $R_3$ is normal. We also prove that the canonical connection of a canonical principal $T^1$-bundle over a Vaisman–Gray manifold $M$ of the class $R_3$ is normal if and only if the Lie vector of the manifold $M$ belongs to the center of the adjoint $K$-algebra.
Keywords: principal toroidal fiber bundle, almost contact structure, curvature tensor. 
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E. E. Ditkovskaya. Curvature identities for principle $T^1$-bundles over almost Hermitian manifolds. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2010), pp. 56-63. http://geodesic.mathdoc.fr/item/IVM_2010_7_a4/

[1] Kirichenko V. F., “Differentsialnaya geometriya glavnykh toroidalnykh rassloenii”, Fundament. i prikl. matem., 6:4 (2000), 1095–1120 | MR | Zbl

[2] Gray A., “Curvature identities for Hermitian and almost Hermitian manifolds”, Tôhoku Math. J., 28:4 (1976), 601–812 | DOI | MR

[3] Kirichenko V. F., Differentsialno-geometricheskie struktury na mnogoobraziyakh, MPGU, M., 2003

[4] Kobayashi S., “Principal fibre bundle with the 1-dimensional toroidal group”, Tôhoku Math. J., 8 (1956), 29–45 | DOI | MR | Zbl

[5] Volkova E. S., O geometrii normalnykh mnogoobrazii killingova tipa, Dep. v VINITI RAN 2111-V96, 1996, 25 pp.

[6] Borisovskii I. P., O geometrii glavnykh $T^{1}$-rassloenii nad pochti ermitovymi mnogoobraziyami, Dep. v VINITI RAN 3729-V97, 1997, 12 pp.

[7] Kirichenko V. F., “Metody obobschennoi ermitovoi geometrii v teorii pochti kontaktnykh mnogoobrazii”, Itogi nauki i tekhniki. Ser. Probl. geom., 18, 1986, 25–71 | MR | Zbl