Given the smooth manifold $M = RP^2 \times \Gamma$ where $RP^2$ is the real projective plane and $\Gamma$ the unitary cylinder, verify that $$Z := \left\{ \big((y_0: y_1: y_2), (x_1, x_2, x_3)\big) \in M :\space y^2_0 = y^2_1 + y^2_2, x_1 = {1\over2}\right\}$$ is an embedded sub-manifold of M.
I'm a little bit confused because if there were no projective, I would have used the result that pre-image of regular value is a sub-manifold with a light heart, but in this case, I was thinking about using slice condition or similar. There exist a "smarter" way to proceed rather than calculations. Thanks a lot!