Curvature identities for principle $T^1$-bundles over almost Hermitian manifolds

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.