I'm working in an earlier edition of John Lee's book on smooth manifolds, and he has a number of problems where he represents a real manifold using complex variables. For instance in chapter 3 problem 5:
Consider $\mathbb{S}^3$ as a subset of $\mathbb{C}^2$ under the usual identification of $\mathbb{C}^2$ with $\mathbb{R}^4$. For each $z = (z^1, z^2) \in \mathbb{S}^3$ define a curve $\gamma_z: \mathbb{R} \to \mathbb{S}^3$ by $$ \gamma_z(t) = (e^{it}z^1, e^{it}z^2).$$
He subsequently asks for us to compute the coordinate representation of $\gamma_z(t)$ using stereographic projection and then also compute $\gamma_z'(t)$ under the same coordinate transformation. I'm really confused about to do this using complex variables when we're working in a real manifold. Out of stubbornness I computed what I thought would be the real representation of $\gamma_z(t)$:
$$ \gamma_x(t) \;\; =\;\; (x^1 \cos t - x^2 \sin t, x^1 \sin t + x^2 \cos t, x^3 \cos t - x^4 \sin t, x^3 \sin t + x^4 \cos t) $$
but this can't possibly be right. Letting $\sigma:\mathbb{S}^3/\{N\} \to \mathbb{R}^3$ be the stereographic projection from the neighborhood omitting the point $N = (0,0,0,1)$, we then obtain
$$ (\sigma\circ \gamma_x)(t) \;\; =\;\; \frac{(x^1 \cos t - x^2 \sin t, x^1 \sin t + x^2 \cos t, x^3 \cos t - x^4 \sin t)}{1 - x^3\sin t - x^4 \cos t} $$
which can't possibly be right since it's not defined at the point $(0,0,1,0)$ for all $t \in \mathbb{R}$.
I'm lost as to how to tackle this kind of problem using complex variables. Thanks in advance!
P.S. For reference, my copy is a Chinese edition but I don't think there's any substantive difference between my copy and the one posted in the link. I definitely do not have the second edition.
Your formulas are fine. The "coordinate representation" of a map from $\mathbb R$ into a manifold is generally defined only on a subset of $\mathbb R$, namely the subset that maps into the domain of the coordinates. In this case, the domain of stereographic coordinates is $U = \mathbb S^3\smallsetminus \{(0,0,0,1)\}$. For any $z$ of the form $z=(0,z^2)$ (or $(0,0,x^3,x^4)$ in real coordinates), there will be infinitely many values of $t$ for which $\gamma_z(t)$ is not in the domain of stereographic coordinates. For example, as you noted, if $z = (0,0,1,0)$, then $\gamma_z(t)\notin U$ when $t$ is an odd multiple of $\pi/2$, so the coordinate representation of $\gamma_z$ is not defined for these values of $t$.