Differential geometry of principal toroidal fiber bundles

We study the differential geometry of principal toroidal fiber bundles of arbitrary rank over a smooth manifold that may be equipped with some additional structure. The explicit calculation is given for the characteristic class of the canonical principal $T^1$-bundle over an almost Hermitian manifold. We study the structure of Riemann–Christoffel tensor and Ricci tensor of the canonical pseudo-Riemannian structure induced on the space of the principal toroidal fiber bundle over a pseudo-Riemannian manifold, and a criterion is found for this structure to be Einstein one. We study the properties of the almost contact metric structure canonically induced on the space of a canonical principal toroidal fiber bundle over an almost Hermitian manifold. The obtained results generalize some known results in this area and allow to build new interesting examples of almost contact metric structures of various classes.