Let $f\colon\mathbb{R}P^3 \to \mathbb{R}$
$$f (x_1, x_2, x_3, x_4) = \frac{x_1^2+x_2^2+x_3^2-x_4^2}{x_1^2+x_2^2+x_3^2+x_4^2}$$
Is $f^{-1}(0)$ a sub-manifold of $\mathbb{R}P^3$?
My approach is to extend $f$ to $F$ on $\mathbb{R}^4$, and check that every vector in $T_{p}\mathbb{R}P^3$ is in $\ker dF_{p}$. ($p$ is a point in $f^{-1}(0)$) However, I think it is difficult to explicitly express a vetor in $T_{p}\mathbb{R}P^3$.