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. 
@article{IVM_2010_7_a4,
     author = {E. E. Ditkovskaya},
     title = {Curvature identities for principle $T^1$-bundles over almost {Hermitian} manifolds},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {56--63},
     publisher = {mathdoc},
     number = {7},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2010_7_a4/}
}
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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/