A torus in $\mathbb{R}^3$ can be defined parametrically by
\begin{aligned} x(\theta ,\varphi )&=(R+r\cos \theta )\cos {\varphi }\\ y(\theta ,\varphi )&=(R+r\cos \theta )\sin {\varphi }\\ z(\theta ,\varphi )&=r\sin \theta \end{aligned} where $\theta, \varphi$ are angles which make a full circle (so that their values start and end at the same point); $R$ is the distance from the center of the tube to the center of the torus; $r$ is the radius of the tube.
Denote the unit $3$-sphere by $\mathbb{S}^3=\lbrace (x,y,z,w)\in\mathbb{R}^4\mid x^2+y^2+z^2+w^2=1\rbrace$. I know some families of tori in the $3$-sphere, like the Clifford tori given by $$\left\lbrace\left(\sqrt{{1+a\over 2}}\cos{\theta}, \sqrt{{1+a\over 2}}\sin{\theta}, \sqrt{{1-a\over 2}}\cos{\phi},\sqrt{{1-a\over 2}}\sin{\phi} \right) \, \mid \, 0 \leq \theta < 2\pi, 0 \leq \phi < 2\pi \right\rbrace$$ or the Hopf tori constructed using the inverse of the Hopf fibration $p:\mathbb{S}^3\to\mathbb{S}^2$ of the $3$-sphere over the $2$-sphere. I believe that both Clifford and Hopf tori are conformal in $\mathbb{S}^3$.
My questions:
- Is it possible at all to find a general parametrization of a torus in $\mathbb{S}^3$ like the one before in the case of $\mathbb{R}^3$?
My guess is that something like $$(R\cos u,R\sin u,P\cos v,P\sin v)$$ where $R$ and $P$ are constants (or maybe they shouldn't be constants) determining the aspect ratio could work, but it looks too simple compared with the $\mathbb{R}^3$ case.
- Is the conformality of the previous families of tori something somehow general in $\mathbb{S}^3$?
The fact that these parametrizations are conformal and that $\mathbb{S}^3$ is the natural space to work in when dealing with conformal problems makes me think that maybe all tori in $\mathbb{S}^3$ could be written using a conformal parametrization.
Your question makes me wonder what do you mean by a torus. Q1: I believe there isn't a general parametrization of tori in $\mathbb S^3$. Indeed I do not see one even in $\mathbb R^3$: A torus in $\mathbb R^3$ is just a submanifold diffeomorphic to $\mathbb S^1 \times \mathbb S^1$. It doesn't have to be parametrized as suggested in the question.
Q2: Agagin it is not clear what you want. If $T\subset \mathbb S^3$ is a torus, then endow the metric $i^* g$ on $T$, where $i :T\to \mathbb S^3$ is the inclusion and $g$ the standard metric on $\mathbb S^3$. Then $(T, i^* g) \to (\mathbb S^3,g)$ is a conformal immersion.
On the other hand, since by the uniformization theorem, $(T, i^*g)$ is conformal to $\mathbb R^2/\Gamma$ for some lattice $\Gamma$, so basically every tori in every manifolds are conformal with respect to some metric.
As a side remark U. Pinkall proved here that every tori $\mathbb R^2/ \Gamma$ can be conformally embedded into $\mathbb S^3$.