consider the map map $\pi $ $: \mathbb{S}^{n} \subseteq \mathbb{R}^{n+1} \rightarrow \mathbb{P}^{n} \mathbb{R}$ obtained by resticting the projection map $\mathbb{R}^{n+1} \setminus \{0 \} \rightarrow \mathbb{P}^{n} \mathbb{R}$. What is the rank of $\pi$ at any given point?
I tried to solve it step by step by plugging in the definitions and I got something messy wich gives the right result (I'm also not sure about the correctness of this messy formula). Now the official solution to this exercise is a bit confusing, so I would like to have suggestions on how you would solve this problem. I would appreciate any complete answer. Thanks.
Consider $e\in S^n$ and $\xi=\pi(e)$. Let $H$ be the tangent plane to $S$ at $e$. It is given by $(x-e)\cdot e=0$. Then $H$ gives local coordinates to $P^n \Bbb{R}$ in a neighborhood of $e$. (Now this depends on how you have defined local coors of projective space?). You may introduce a base in $H$ and coordinates, but that tends to obscure the idea.
$H$ also provide local coors for $S$ at $e$: The map $h(x) = x/|x|$ maps $H$ onto a nghb of $e$ in $S$ and the inverse is $h^{-1}(y)=y/(e\cdot y)$. So locally $S$ and $P{\Bbb R}^n$ are diffeomorphic in a nghb of $e$.