Let $g:(x,y,z,w)\mapsto x^2+y^2+z^2-w^2$. Then by the regular value theorem, the set $g^{-1}(-1)$ is a submanifold of $\mathbb{R}^4$ of codimension 1.
The question is, if $f:(x,y,z,w)\mapsto w$, then what are the critical values of the restricted map $f|_{g^{-1}(-1)}$?
The map $f$ has the Jacobian $[0,0,0,1]$, so is it not of full rank everywhere (hence no critical value of the restriction)?
What if the map was $f:(x,y,z,w)\mapsto w^2$? In that case, the Jacobian fails to be of full rank precisely when $w=0$. But $g^{-1}(-1)$ has no point where $w=0$, hence no critical value here as well?
it is a hyperboloid of two sheets. There are two critical points of your $w$ which occur when $w = \pm 1$ and $x=y=z=0$