Periodicity of the asymptotic curves on flat tori in \(S^ 3\) (Q1101706)

In this paper, we establish a method of construction of flat tori in S 3. Every flat torus in S 3 is constructed by this method. Consequently, we see that the problem of the classification of flat tori in S 3 reduces completely to that of the classification of pairs of two closed curves on S 2 which satisfy certain geometric inequality. This result is applied to prove the periodicity of the asymptotic curves on flat tori in S 3.