Hopf Tori in $S^3$ - MathOverflow

Hopf Tori in $S^3$

2012-01-16T12:31:07Z

By means of the Hopf fibration $\pi: S^3 \rightarrow S^2$, U. Pinkall showed that every compact Riemannian surface of genus one can be conformally embedded in $S^3$.

More precisely: Let $p$ be a closed curve on $S^2$ of length $L$. Lifting it to $S^3$ yields a torus isometric to $R^2 / \Gamma$, with $\Gamma$ generated by $(2\pi, 0)$ and $(A/2, L/2)$, where $A$ is the area enclosed by $p$. (This is Proposition 1 in Pinkall's paper.)

Now it's claimed that if you lift a great-circle you should get the Clifford-torus. The above proposition then yields that the Clifford-torus is isometric to $R^2 / \Gamma _1$, with $\Gamma _1$ generated by $(2\pi, 0)$ and $(\pi, \pi)$. The usual definition of the Clifford torus is $R^2 / \Gamma _c$ with $\Gamma _c$ generated by $(2\pi, 0 )$ and $(0, 2 \pi)$.

Who know's how this fits together?

Answer by BS for Hopf Tori in $S^3$

2012-01-16T13:08:26Z

If $T_1,T_2$ are any 1-parameter subgroups of $S^3=SU(2)$ viewed as the unit sphere of quaternions, Clifford tori are realized as $T_1.q.T_2$, $q\in S^3$.

But this is isometric to the quotient of $T_1\times T_2$ by $(-1,-1)$, hence the $(\pi,\pi)$ in the lattice $\Gamma_1$, since $T_i\simeq\mathbb{R}/2\pi\mathbb{Z}$.

Answer by Vitali Kapovitch for Hopf Tori in $S^3$

2012-01-16T14:03:22Z

Your definition of the Clifford torus is off. The usual definition of the Clifford torus is the set $(z_1,z_2)\in\mathbb C^2$ in the unit sphere $|z_1|^2+|z_2|^2=1$ with $|z_1|^2=|z_2|^2=\frac 1 2$. This is a square torus isometric to $\mathbb R^2/\Gamma_c$ with $\Gamma_c$ generated by $(2\pi/\sqrt 2, 0), (0, 2\pi/\sqrt 2)$ (not $(2\pi, 0), (0, 2\pi)$) which is isometric to $\mathbb R^2/\Gamma_1$ by a $\pi/4$ rotation.