  We present traps with toroidal $(T^{2})$ and ring-shaped topologies, based on
adiabatic potentials for radio-frequency dressed Zeeman states in a ring-shaped
magnetic quadrupole field. Simple adjustment of the radio-frequency fields
provides versatile possibilities for dynamical parameter tuning, topology
change, and controlled potential perturbation. We show how to induce toroidal
and poloidal rotations, and demonstrate the feasibility of preparing degenerate
quantum gases with reduced dimensionality and periodic boundary conditions. The
great level of dynamical and even state dependent control is useful for atom
interferometry.
