All actions of $SO(3)$ on $S^2$

Question

There are two obvious (smooth left) actions of $\mathrm{SO}(3)$ on $S^2$. There is the standard action by which $\mathrm{SO}(3)$ acts by 3D rotations on the standard embedding of $S^2$ in $\mathbb{R}^3$, and there is the trivial action in which $\mathrm{SO}(3)$ leaves the points of $S^2$ unchanged. It feels like these are the only two options. Is that intuition correct?

Answer 1

No, there are other smooth actions:

Fix $g\in\mathrm{SO}(3)$ and denote by $(h,x)\mapsto h.x$ the standard action. Then

$$ \mathrm{SO}(3) \times\mathbb S^2\to\mathbb S^2,\;\;(h,x)\mapsto ghg^{-1}.x $$ is also a smooth left action, which is not the trivial action and in general different from the standard one.