A rigidity theorem for the Clifford tori in $S^3$

Abstract:

Let $S^3$ be the unit hypersphere in the 4-dimensional Euclidean space $\Bbb R^4$ defined by $\sum _{i=1}^4 x_i^2=1$. For each $\theta$ with $0<\theta <\pi /2$, we denote by $M_\theta$ the Clifford torus in $S^3$ given by the equations $x_1^2+x_2^2=\cos ^2\theta$ and $x_3^2+x_4^2= \sin ^2\theta$. The Clifford torus $M_\theta$ is a flat Riemannian manifold equipped with the metric induced by the inclusion map $i_\theta \colon M_\theta \to S^3$. In this note we prove the following rigidity theorem: If $f\colon M_\theta \to S^3$ is an isometric embedding, then there exists an isometry $A$ of $S^3$ such that $f=A\circ i_\theta$. We also show no flat torus with the intrinsic diameter $\le \pi$ is embeddable in $S^3$ except for a Clifford torus.