Riemann sphere and Fundamental theorem of algebra

Question

Recently, I saw the following proof of the Fundamental theorem of algebra through the Gauss-Bonnet theorem. https://arxiv.org/pdf/1106.0924.pdf
In here, $\mathbb{C} \cup \{\infty\}$ is identified to $\mathbb{S}^2$. Now, I have a question. $\mathbb{C} \cup \{\infty\}$ is homeomorphic to $\mathbb{S}^2$. However, is it diffeomorphic to $\mathbb{S}^2$? To use the Gauss bonnet theorem, $\mathbb{C} \cup \{\infty\}$ should be the smooth manifold. I have a trouble with the differentiability at $\infty$.

Answer 1

The sphere $S^2=\mathbb{C}\cup\{\infty\}$ becomes a differentiable manifold once we've put a differentiable structure in it. We can do it with two charts:

1. The identity from $\mathbb{C}(\subset S^2)$ into $\mathbb C$;
2. The map $\varphi\colon(\mathbb{C}\setminus\{0\})\cup\{\infty\}\longrightarrow\mathbb C$ defined by$$\varphi(z)=\begin{cases}\frac1z&\text{ if }z\neq\infty\\0&\text{ otherwise.}\end{cases}$$