  We study a quasispin-$1/2$ Bose-Einstein condensate with synthetically
generated spin-orbit coupling in a toroidal trap, and show that the system has
a rich variety of ground and metastable states. As the central hole region
increases, i.e., the potential changes from harmonic-like to ring-like, the
condensate exhibits a variety of structures, such as triangular stripes,
flower-petal patterns, and counter-circling states. We also show that the
rotating systems have exotic vortex configurations. In the limit of a quasi-one
dimensional ring, the quantum many-body ground state is obtained, which is
found to be the fragmented condensate.
