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Problem:

In the construction of the Riemann sphere, we begin with the sphere $\mathbb{S}^2$ with two charts:

  1. the stereographic projection $\sigma_N : \mathbb{S}^2 \setminus \{N\} \to \mathbb{R}^2 \cong \mathbb{C}$ from the North pole, $N$, given by $$ \sigma_N (x_1, x_2, x_3) := \frac{(x_1, x_2)}{1-x_3}, $$

  2. the stereographic projection $\sigma_S : \mathbb{S}^2 \setminus \{S\} \to \mathbb{R}^2 \cong \mathbb{C}$ from the South pole, $S$, given by $$ \sigma_S (x_1, x_2, x_3) := \frac{(x_1, x_2)}{1+x_3}. $$

Question:

How does one show that the transition function of the two charts is: $$\sigma_1 \circ \sigma_0^{-1} (z) = z^{-1}$$

Remark:

By elementary calculations we see that: $$\sigma_S(x_1,x_2,x_3)=\sigma_N(x_1,x_2,-x_3)$$ $$\sigma_N(x_1,x_2,x_3)=\sigma_S(x_1,x_2,-x_3)$$

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1 Answer 1

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Let the stereographic projection $\sigma_0 : \mathbb{S}^2 \setminus \{N\} \to \mathbb{C}$ from the North pole, $N$, be given by $$ \sigma_0 (x_1, x_2, x_3) := \frac{x_1 + ix_2}{1-x_3}, $$ and the stereographic projection $\sigma_1 : \mathbb{S}^2 \setminus \{S\} \to \mathbb{R}^2 \cong \mathbb{C}$ from the North pole, $S$, be given by $$ \sigma_1 (x_1, x_2, x_3) := \frac{x_1 - ix_2}{1+x_3} $$ (notice how the orientation has been reversed via complex conjugation). The inverse of $\sigma_0$ is given by $$ \sigma_0^{-1} (z) = \sigma_0^{-1} (u+iv) = \frac{(2u,2v,|z|^2-1)}{|z|^2+1}. $$ The computation now shows that indeed the transition function is $$ \sigma_1 \circ \sigma_0^{-1} (z) = \frac{1}{z}. $$

Notice moreover that $$ \sigma_1^{-1}(z) = \frac{(2u,2v,1-|z|^2)}{|z|^2+1}, $$ so a computation shows that $$ \sigma_0 \circ \sigma_1^{-1} (z) = \frac{1}{z^\ast}. $$

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2  
Shouldn't $\sigma_0\circ\sigma_1^{-1}(z)=\frac{1}{z}$ too? –  Freeze_S Jun 10 '14 at 9:03
    
Something is wrong with a sign somewhere. If $\sigma_0$ and $\sigma_1$ are really just the stereographic projections from the poles, then $\sigma_0\circ \sigma_1^{-1}$ should just be inversion in the unit circle, which is $z\mapsto 1/\overline{z}$, but then $\sigma_1\circ\sigma_0^{-1}$ should be too, since $\sigma_1\circ\sigma_0^{-1} = (\sigma_0\circ\sigma_1^{-1})^{-1}$. You wanted to give $\sigma_1$ the orientation-reversal so that the transition maps would be holomorphic; but then both composed transition maps should be $z\mapsto 1/z$. –  Ben Blum-Smith Jan 3 at 21:42

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