  We investigated the nematic ordering on a torus by means of analytic method
and the method of simulated annealing, the Frank free energy, both in the
standard form and covariant form, were used in the study. The defect free state
was found to be the ground state in both cases. However, in the case of the
standard model, there are two kinds of defective free ordering and a transition
between the two occurs at a critical value of radius ratio $k=\frac{r}{R}$. The
first one is $\theta=0$ in the small $k$ regime and the second one is a
variable $\theta$ with position of the torus. In the case of the covariant
model the ground state is confirmed to be the infinitely degenerate of $\theta$
equals to a random constant. The states with defects are the excited states,
where the pairs of defects excited and, duo to the barrier between positive and
negative defects, have pretty long life. The behavior of the defect state
basically the same for both of the two models.
