Compute pullback of differential form Let $\omega = pdq-qdp-rds+sdr$ be a $1$-form on $\mathbb{R}^4$. Identify $\mathbb{R}^4$ with $\mathbb{C}^2$ as follows: $(p,q,r,s) \to (p+iq,r+is)$. Define an embedding $\iota\colon S^2 \to \mathbb{C}^2$ by the formula $\iota(x,y,z)=(x+1,y+iz)$. For each $[a:b] \in \mathbb{C}P^1$ the corresponding complex line $\{(au,bu) | u \in \mathbb{C}\}$ intersects $\iota(S^2)$ in exactly one point $f(a,b)=\iota(x(a,b),y(a,b),z(a,b))$. Compute the $1$-form $\nu = f^*\iota^*\omega$ on $\mathbb{C}P^1$.
I understand what is pullback in general and able to compute it in simple cases, but this huge condition really confused me. Can anyone show me how to do it or at least clarify sequencing on what to do with all this variables substitutions step by step? If it is not legitimate to provide full solutions here (I'm newbie on this forum and really don't know), it even will be enough to show $\iota^*\omega$ and further calculations remain to me.
 A: Let me show you how you can compute $\iota^{*}(\omega)$ (which is admittedly the easier part). We think of $S^2$ as sitting inside $\mathbb{R}^3$ and have a map $\iota(x,y,z) = (x,1,y,z) = (p,q,r,s)$ (where by purpose I have ignored the identification of $\mathbb{R}^4$ with $\mathbb{C}^2$ so that we won't be confused by the presence of complex numbers). Written explicitly, we have
$$ p = x, q = 1, r = y, s = z $$
and so
$$ \iota^{*}(pdq - qdp - rds + sdr) = x d(1) - 1 dx - ydz + zdy = -dx - ydz + zdy.$$
Now, consider the coordinate patch $a = u + iv \mapsto [a:1]$ on $\mathbb{C}^2$. By compositing it with $f$, we should get a map $f \colon \mathbb{C} \rightarrow S^2$ which has the form (ignoring again the complex identification)
$$ (u, v) \mapsto (x(u,v), y(u,v), z(u,v)). $$
By repeating the procedure above (plugging $x(u,v),y(u,v),z(u,v)$ in $-dx - ydz + zdy$ and calculating the result), you'll get the formula for the total pullback on the subset $\{ [a : b] \, | \, a,b \in \mathbb{C}, b \neq 0 \}$ written in the coordinate chart $a \mapsto [a:1]$. Similarly, you can use a coordinate patch $b \mapsto [1 : b]$ to get the total pullback on the subset $\{ [a : b] \, | \, a, b \in \mathbb{C}, a \neq 0 \}$ written in the coordinate chart $b \mapsto [1 : b]$.
