Complex Geometry/Holomorphic manifolds

Definition (holomorphic manifold):

A holomorphic manifold is a differentiable manifold whose transition maps are holomorphic.

Example (Riemann sphere):

Consider the set $\mathbb {C} \cup \{\infty \}$

Theorem (Riemann's extension theorem):

Let $M$ be a complex manifold, let $f:M\to \mathbb {C}$ be holomorphic, and let $g:M\setminus Z(f)\to \mathbb {C}$ be holomorphic, such that for all $p\in Z(f)$ and all $\epsilon >0$, the function $g$ is bounded on $B_{\epsilon }(p)\cap Z(f)^{c}$. Then there exists a unique function $G:M\to \mathbb {C}$ that extends $g$.

Exercises

1. Consider the stereographic projection from a sphere to the complex plane, where the sphere is of radius $1$ and tangent to the zero point of the complex plane. Prove that the function $z\mapsto 1/z$ corresponds to a reflection of the sphere about the equator.