How can I prove that the set operators with rank $k$ is a manifold? More precisaly I would like prove that the set $\{ P\in L(\mathbb{R}^{m},\mathbb{R}^{m}): P^{2}=P\quad and \quad rank(P)=k\}$ is a manifold of class $C^{\infty}$ and dimension $k(m-k)$ in $\mathbb{R}^{m^{2}}$. This is a question of Elon's lima book.
Thank you very much.