Does projectivizing always fix problems at infinity? (Or, am I making a mistake somewhere?)

This question is motivated by the following homework problem. I'm trying to explicitly compute the homeomorphism $f:S^2 \rightarrow \mathbb{CP}^1$ by using stereographic projection and considering $\mathbb{CP}^1 = \mathbb{C}\cup {\infty}$. I'll want to prove that this is an isometry, where $S^2$ has the standard angle metric and $\mathbb{CP}^1$ has the Fubini-Study metric given by $d(\overline{x},\overline{y})=2\cos^{-1}|(x,y)|$, where $x,y\in \mathbb{C}^2$ are unit vectors (and presumably $(-,-)$ is the usual Hermitian inner product). Later, I'll use this to explicitly compute the Lie group homomorphism $U(2)\rightarrow SO(3)$.

My stereographic projection is from the north pole, takes the equator to the unit circle, and puts the south pole at the origin. What I've gotten so far is that for $z\not= 1$, \begin{equation*} f(x,y,z)=\left( \frac{x}{1-z} , \frac{y}{1-z} \right) = \frac{x+iy}{1-z} = [x+iy : 1-z ], \end{equation*} where these are coordinates in $\mathbb{R}^2$, $\mathbb{C}$, and $\mathbb{C}\subseteq \mathbb{CP}^1$ respectively. This is troublesome, because philosophically I'd expect that I should be able to define this for $(x,y,z)\not= (0,0,1)$ and then end up with a function to projective space that extends continuously over the north pole; that's sort of the point of projective space, to make $\infty$ into just another point. However, it is not immediately obvious that this works, although luckily \begin{equation*} \left| \frac{x+iy}{1-z} \right| = \sqrt{ \frac{|x+iy|^2}{(1-z)^2} } = \sqrt{ \frac{1-z^2}{(1-z)^2}}, \end{equation*} and the limit of this expression as $z\rightarrow 1^-$ is indeed $\infty$.

So, fair enough. This ends up extending to a continuous function after all. But: Am I wrong in my philosophical understanding of projective space?

(For what it's worth, I tried using my calculations to verify that $f$ is an isometry, and it didn't look like it was going to work out. So maybe I really am just doing something wrong.)

Solution 1:

One complication in your situation is that you are mixing real and complex coordinates.
If you were considering a map from a complex curve to $\mathbb{CP}^1$, then the kind of computation you are trying to make would work out more straightforwardly.

Because you are looking at a map of real analytic manifolds, not complex analytic ones (concretely, you are working with the variables $x,y,z$, which are real coordinates), the point of view you have adopted is perhaps not quite as natural. Nevertheless, it can be made to work, as follows: $$[x+iy:1-z] \text{ (which is where you finished) } = [x^2 + y^2: (1-z)(x - i y)]$$ $$ = [ 1 - z^2: (1-z)(x-iy)] = [1+z:(x-iy)].$$ This rewriting of your map to $\mathbb{CP}^1$ is now well-defined in a neighbourhood of $(0,0,1)$ on the sphere. (The fact that I introduced a complex conjugate of $x + i y$ to facilitate the computation is related to the real vs. complex issue mentioned above. This is also essentially the same computation you made to check that your map tends to $\infty$ as $z \to 1$, just rewritten in homogeneous coordinates.)