Construction of flat tori in the 3-sphere and its applications (Q2752880)

In the present paper the author surveys methods to construct flat tori in \(S^3\) and explains some classification result.NEWLINENEWLINELawson observed that, if \(p: S^3 \to S^2\) is the Hopf fibration and \(\gamma\) is a curve on \(S^2\), then \(p^{-1} (\gamma)\) is a flat surface, the Hopf torus. On the other hand, there is an older method, which goes back to Bianchi, to construct flat surfaces in \(S^3\), using the group structure of \(S^3\). Namely, if \(\alpha\) and \(\beta\) are curves in \(S^3\) with torsion \(1\) and \(-1\) respectively, then \(F(u,v):=\alpha (u) \beta (v)\) is a flat surface. The author obtains a method to construct closed curves with torsion \(\pm 1\). This allows him to establish a method for constructing all flat tori isometrically immersed in \(S^3\). As a consequence, he proves that every embedded flat torus is invariant under the antipodal map of \(S^3\), which implies a rigidity theorem for the Clifford torus. Furthermore, the author obtains the classification of undeformable flat tori in \(S^3\).NEWLINENEWLINEFor the entire collection see [Zbl 0960.00049].