Let $P^{n-1}(\mathbb{R})$ the real projective space of dimension $n-1$. Consider the set
$$B=\{(x,y)\in\mathbb{R}^n \times P^{n-1}(\mathbb{R}) / x=(x_1,..,x_N), y=[y_1;..;y_N], x_iy_j=x_jy_i \hspace{0.2cm} \forall i,j =1-n\}$$
The problem is to show that $(B,i)$, being $i$ the inclusion map, is a submanifold of the manifold $\mathbb{R}^n \times P^{n-1}(\mathbb{R}) $. I have proved that $B$ is well defined and that $di_p$ is injective $\forall p \in B$ (and it's clear that $i$ is injective). But how can I show that $B$ is a submanifold? I'm not sure how to prove that it has a differential structure inside $\mathbb{R}^n \times P^{n-1}(\mathbb{R})$.