Embedded flat tori in the unit 3-sphere

Abstract

Let $S^{3}$ be the unit hypersphere in the 4-dimensional Euclidean space $E^{4}$ given by $\Sigma_{i=1}^{4}x_{i}^{2}=1$ . For each $\theta$ with $0<\theta<\pi/2$ , we consider a surface $M_{\theta}$ in $S^{3}$ defined by