I am reading the script http://www3.math.tu-berlin.de/geometrie/Lehre/WS16/DGII/script.pdf and I want to prove exercise 9 where you need to show that $G_1(\Bbb R^3) \subset sym(3)$ is a submanifold diffeomorphic to $ \Bbb RP^2 $.
The Grassmanian of k-plane in $ R^n $ is defined as follows
$ G_k(\Bbb R^n):= \{ p\in \Bbb R^{n \times n}|p^*=p , p^2=p, tr(p)=k \}$
and there is a theorem that says $ G_k(\Bbb R^n)$ is a submanifold of $\Bbb R^{n \times n}$ of dimension $k(n-k)$.
I don't really see how I can show it's a submanifold of $sym(3)$ and how it's diffeomorphic to $\Bbb RP^2$ .