The exercise says the following:
Show that a projective line is a submanifold of $\mathbb{RP}^{n}$ diffeomorphic to $\mathbb{RP}^{1}$.
I think this is a consequence of the following result
- if $q:\mathbb{R}^{n+1}\setminus \{0\}\to \mathbb{RP}^{n}$ be the usual quotient map. For any subspace $U\subset \mathbb{R}^{n+1}$, $q(U\setminus \{0\})$ is a submanifold of $\mathbb{RP}^{n}$ of dimension equal to $dim(U) − 1$.
for the previous result if $dim(U)=2$, then $q(U\setminus \{0\})$ is a projective line and a submanifold of $\mathbb{RP}^{n}$. I have shown that the $q$ function is smooth, but I don't know how to show that the image below $q$ is a submanifold. Thanks for the help.