Let $f : S^3 → S^2$ by $f(x_0, x_1, x_2, x_3) = (x_0^2+x_1^2-x_2^2-x_3^2, 2x_0x_3+2x_1x_2, 2x_1x_3-2x_0x_2)$
Show that $f^{-1}\{y\} \subseteq S^3$ is a circle for all $y \in S^2$.
Can you help me in this exercise ? I am learning immersion, embedding, submersion and I have this exercise that might be related where I don't get how to start the proof.
Strong hint
I suspect that if you treat $S^3$ as sitting in $\Bbb C^2$, and so each point of $S^3$ is a pair $(z, w)$ with $|z|^2 + |w|^2 = 1$, then your function $f$ could be described by $$ (z, w) \mapsto z/w $$ where the quotient takes place in the extended complex plane, which is then mapped to the unit sphere via some sort of stereographic projection.
If that's the case, then this is just the Hopf map in coordinates.
Let's look at $f^{-1}(1)$; this is the set of points $(z, z)$ where $2|z|^2 = 1$, which is evidently a circle.
A little more work lets you conclude the same thing even when the target is some other complex value $c$.