Is there a surjective $C^{\omega}$ map from the 2-sphere to the 2-torus?
More explicitly, is there a surjective real-analytic mapping $f \colon S^2 \to T^2$ where
$$S^2 = \{x \in \mathbb{R}^3 : \|x\| = 1\}, \quad T^2 = \{x \in \mathbb{R}^4 : x_1^2 + x_2^2 = x_3^3 + x_4^2 = 1\}?$$
There are $C^{\infty}$ surjective maps from the 2-sphere to the 2-torus, but all of the constructions I know of are not analytic.
Attempt: Suppose such $f$ exists, and consider the projection $P(x_1, x_2, x_3, x_4) = (x_1, x_3)$. Then $P \circ f \colon S^2 \to \mathbb{R}^2$ is an analytic surjective map from the sphere to the closed square with vertices $(\pm 1, \pm 1)$. I would guess that no such map exists.
Best Answer
There is a nice map from the plane $\mathbb{R}^2$ to the Clifford torus $T^2$ given by
$$(\alpha,\beta)\mapsto(\cos\alpha,\sin\alpha,\cos\beta,\sin\beta).$$
(You can also think of this in $\mathbb{C}^2$ as $(e^{i\alpha},e^{i\beta})$ if desired.) The trig functions are of course real-analytic. This is doubly periodic, and in particular remains onto restricted to any domain containing $[-\pi,\pi]^2$.
There is a nice map from the unit sphere to the unit disk given by projecting, given by
$$ (x,y,z)\mapsto(x,y) $$
If we expand the unit disk by a factor of $\sqrt{2}\pi$ it will just be enough to contain $[-\pi,\pi]^2$. Therefore,
$$ f(x,y,z)=\Big(\cos(\sqrt{2}\pi x),\,\sin(\sqrt{2}\pi x),\,\cos(\sqrt{2}\pi y),\,\sin(\sqrt{2}\pi y)\Big) $$
is an onto, real-analytic map from the unit sphere $S^2$ to the Clifort torus $T^2$.