Pick $ p \in S^{n-1} \subset \mathbb{R}^n$ and consider the map $\pi: SO_n \to S^{n-1}$ by $A \to A(p)$. Show that this map is a smooth submersion. For $ q \in S^{n-1}$, describe the pre-image.
For context: I am currently an undergraduate in a first course on smooth manifolds. I know how to show the map is smooth by just restricting the same map defined on $GL_n$ to the embedded sub manifold $SO_n$. However, I'm having trouble showing that this map is a submersion. My first thought is to try to use the velocity vector of some curve. Given $A \in SO_n, v \in T_{A(p)}S^{n-1}$ to define some smooth map $\gamma: J \to SO_n$ where $J \subseteq \mathbb{R}$ is an interval containing $0$ with $\pi \circ \gamma (0) = A(p)$ and $(\pi \circ \gamma)'(0) = v$. But, writing down an $SO_n$ valued map is not so easy for me.
My second instinct is to try to use a smooth local section. But, again, not quite sure how one does this. What is the right approach?
Looking forward in the book, I think that this can be dealt with rather easily using Lie group actions. But, we haven't covered those, so I am unfamiliar with that tool.
You have to show that for each $A$ the differential of $\pi$ at $A$ is surjective. Set $q=A(p)$. The dimension of the kernel of $d\pi_A$ is the dimension of the stabilzer of $q$ which is $(n-1)(n-2)/2$, thus the dimension of the image of $d\pi_A$ is $n(n-1)/2-(n-1)(n-2)/2=n-1$, thus it is a submersion.