Showing that $SO(2)$ is diffeomorphic(not just isomorphic as groups) to the circle

Question

This is an exercise at the last part of which I am stuck. The obvious diffeomorphisms must be $e^{i\theta} \to A$ and its inverse map. However I am stuck at the part that the inverse map $A \to e^{i\theta}$ is smooth. I cannot figure out what the charts of $SO(2)$ look like... Or is there some more efficient way to make use of? Could anyone please help me?

Answer 1

You can use the fact that $A\mapsto e^{i\theta}$ is the restriction to $SO(2)$ of the map $f:M_2(\Bbb R)\to \Bbb C$ $$A\mapsto a_{1,1}+ia_{2,1}$$ which is smooth (in fact linear). Then you conclude using the fact that $SO(2)\hookrightarrow M_2(\Bbb R)$ is smooth as $SO(2)$ is a submanifold of $M_2(\Bbb R)$.

Answer 2

The diffeomorphism is the map$$\begin{array}{cccl}SO(2)&\longrightarrow&S^1&\\\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}&\mapsto &e^{i\theta}&(=\cos\theta+i\sin\theta).\end{array}$$