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Let $\mathbf r =(w,x,y,z)$ be a unit vector on $\mathbb R^4$. If I am not mistaken, any point on the unit $3$-sphere can be parametrized as

$\left(\begin{matrix}w & x\\ y & z \end{matrix}\right)=\left(\begin{matrix}\cos \theta_1 & -\sin \theta_1\\ \sin \theta_1 & \cos\theta_1 \end{matrix}\right).\left(\begin{matrix}\cos\theta_2 & 0\\ 0 & \sin\theta_2 \end{matrix}\right).\left(\begin{matrix}\cos \theta_3 & \sin \theta_3\\ -\sin \theta_3 & \cos\theta_3 \end{matrix}\right)$,

with $\,\,0\leq\theta_1,\theta_3\leq\frac{\pi}{4}$, $\,\,0\leq\theta_2\leq 2\pi$.

If we keep $\theta_2$ fixed, what is the resulting 2-dimensional manifold which is embedded in this $3$-sphere? I know that keeping $\theta_2$ fixed, breaks the $O(4)$ symmetry of the 3-sphere to $O(2)\times O(2)$. I am wondering, if the topology of this 2-manifold is something well-known like a $2$-sphere, $2$-torus or Klein bottle?

Edit

After reading Kyle Miller's comment, I think, a better way to parametrize the $3$-sphere is by taking the ranges $\,\,0\leq\theta_1,\theta_3\leq2\pi$, $\,\,-\frac{\pi}{4}\leq\theta_2\leq \frac{\pi}{4}$.

user713320
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    I believe you need $0\leq \theta_1,\theta_3\leq 2\pi$ here (based on some graphs I made of a stereographic projection of the surfaces), and the surfaces you get are tori like the first illustration here with circles when $\cos\theta_2=\pm\sin\theta_2$. This is consistent with the $O(2)\times O(2)$ symmetry. – Kyle Miller Aug 01 '21 at 02:54
  • Thank you very much for your comment and the link! The link has helped me to understand the 3-sphere better. – user713320 Aug 11 '21 at 04:02
  • I understand that $\theta_1$ and $\theta_3$ being in the range $0$ to $2\pi$ is perhaps a better choice. I have edited the question accordingly. But this also leads to every point in the 3-sphere covered twice, i.e. we need either $\theta_1$ or $\theta_3$ in $(0,2\pi)$ and the other in $(0,\pi)$ to cover the whole 3-sphere. I can see that $\cos \theta_2=\pm \sin \theta_2$ leads to two circles. But I can't convince myself that the other cases are tori. How can I show that they are tori (if they indeed are)? Naively, they appear more like 2-spheres given the ranges of $\theta_1$ and $\theta_3$. – user713320 Aug 11 '21 at 04:05

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