Determining the genus of the compact orientable surface $S =\{[x_0, x_1, x_2, x_3]\in \Bbb RP^3 : x^2_0+ x^2_1- x^2_2-x^2_3=0\}$

Question

Consider the subset $S =\{[x_0, x_1, x_2, x_3]\in \Bbb RP^3 : x^2_0+ x^2_1-x^2_2-x^2_3=0\}$ of $\Bbb RP^3$. Clearly $S$ is an embedded submanifold of $\Bbb RP^3$ of codimension $1$, so it is a compact orientable (since $\Bbb RP^3$ is compact and orienatble) surface. Hence it must be diffeomorphic to either $S^2$ or a connected sum of tori. I want to determine its genus.

Answer 1

HINT: Can you make a linear change of coordinates in $\Bbb R^4$ so that the equation becomes $$y_0y_2+y_1y_3=0?$$

Answer 2

Not a solution, but a hint to get you thinking more in the right direction.

"Clearly is an embedded submanifold of $\Bbb RP^3$ of codimension 1, so it is a compact orientable (since ℝ3 is compact and orienatble) surface."

I think that perhaps your approach went off the rails here, because $\Bbb R P^2$ is an embedded submanifold of $\Bbb RP^3$ of codimension 1, but is not orientable. ($\Bbb RP^2$ can, for instance, be seen as the subset where $x_3 = 0$.)