Submanifold of real projective space

Question

Would you like to tell me how to prove that $\{[x_{0}:x_{1}:x_{2}]: x_{0}x_{1} + x_{2}^{2} = 0 \} \subset \mathbb{R}\mathbb{P}^{2}$ is a submanifold (of $\mathbb{R}\mathbb{P}^{2}$)?

Answer 1

I believe you want to say $x_0x_1+x_2^2=0$. Let $V$ be that subset. Consider $U_0=\{[x_0,x_1,x_2],x_0\neq 0\}$ define $f([x_0,x_1,x_2])={x_1\over x_0}+{x_2^2\over x_0^2}$ is a submersion on $U_0$ this shows that $V\cap U_0$ is a submanifold.