Let $\pi$ be the canonical projection from $\mathbb{R}^{n+1}/\{0\}$ to $\mathbb{P}^n(\mathbb{R})$. Given a $k$-form $\alpha$ on $\mathbb{R}^{n+1}/\{0\}$ find necessary and sufficient conditions such that there exists a $k$-form $\beta$ on $\mathbb{P}^n(\mathbb{R})$ such that $\alpha = \pi^* \beta$.
What I have done: In order to let $\alpha$ be the pullback we should check that $\alpha$ remains the same along lines. Thus for example if $\alpha = f(x,y)dxdy$ we must have
$f(x,y)dxdy = f(\lambda x, \lambda y)d(\lambda x)d(\lambda y) = \lambda^2f(\lambda x, \lambda y)dxdy$.