I have to solve this one:
"Prove that the application $\psi:\mathbb{R}^{n+1}\setminus\{0\}\rightarrow S^n$, $x\mapsto\frac{x}{\|x\|}$ is a submersion".
My idea is to use that "composition of submersion is a submersion", given the diagram, with $\psi=\pi^{-1}\circ f$
\begin{array}{ccc} & \psi & \\ \mathbb{R}^{n+1}\setminus\{0\} & \rightarrow & S^n \\ f \searrow & & \swarrow \pi \\ & \mathbb{P}^n(\mathbb{R}) & \end{array}
I proved that $f$ is a submersion. Moreover I know that $\pi$ is a local diffeomprphism, so $\pi_{*p}:T_p S^n\rightarrow T_{\pi(p)}\mathbb{P}^n(R)$ is a isomorphism for all $p\in S^n$ and so it is injective and surjective. In conclusion $\pi$ is an immersion and a submersion.
Now the thing is: can I say that $\pi^{-1}$ is a submersion too? Can I say that $\pi^{-1}$ is a local diffeomorphism?
Thank you so much