Action of $S^1$ as a Lie Group on $S^2$ by "rotation"

Question

I'm studying Lie Groups and their actions on manifolds and I was looking for a "concrete" example of an action. All the stuff about $GL(n;\mathbb{R})$ acting on $\mathbb{R}^n$ is ok but I can't visualise it well, so I thought of an action by the Lie group $S^1$ on $S^2$ by "rotation" in this sense: every element of $S^1$ can be identified by and angle $\alpha$ given by the anti-clockwise rotation around the center $(0,0)$; in the same way we can parametrize $S^2$ using latitude and colatitude. The action that I came up with is simply $$ \begin{gather} S^1 \times S^2 \longrightarrow S^2 \\ (\alpha,(\theta, \varphi))\longmapsto (\alpha + \theta, \varphi) \end{gather} $$ which "spins" $S^2$ around the vertical axis. Is this actually an example of a Lie group acting on $S^2$? Are there any other more "concrete" examples I can look at?

Answer 1

Indeed, that is an example of an action. Notice that there is another way to write the action without coordinates: if you think of $\mathbb{S}^{1}$ as the set of complex numbers of modulus one, the action $\mathbb{S}^{1}\curvearrowright \mathbb{S}^{2}$ is given by $z\cdot (w,t)=(zw,t)$.

If you want to see some other examples of Lie group actions, here are some:

1. Take $\mathbb{RP}^{2}$ to be the quotient of the $2$-sphere by the relation $x\sim -x$. The action $\mathbb{S}^{1}\curvearrowright \mathbb{S}^{2}$ descends to an action $\mathbb{S}^{1}\curvearrowright \mathbb{RP}^{2}$. Try to think on what the orbits are in this case!

2. Consider the Lie group $G=SO(2)\times \mathbb{R}$. This group acts on $\mathbb{R}^{3}$ by letting $(A,t)\cdot (x,y,z)=(A(x,y),z+t)$. Again, which are the orbits for this action?

Hope this helps!