We investigate models of stationary, self-gravitating, perfect-fluid tori (disks) rotating around black holes, focusing on geometric properties of spacetime. The models are constructed within the general-relativistic hydrodynamics, assuming differential (Keplerian) rotation of the fluid. We discuss a parametric bifurcation occurring in the solution space, different possible configurations of ergoregions (including toroidal ergoregions associated with the tori), and nonmonotonicity of the circumferential radius, as well as the impact of the torus gravity on the location of the innermost stable circular orbit.