Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 3 / 2F (Geometry)

LetBy the regular level set theorem, $F^{-1}(0)$ is a smooth surface wherever $\nabla F\ne 0$. A critical point on the level set would satisfyThese equations imply $4a^3+27b^2=0$. Therefore

If $4a^3+27b^2=0$ and $(a,b)\ne (0,0)$, there is a nonzero real number $r$ such thatThen $(0,0,r)$ is singular andFor $r>0$, the singular point is locally isolated because all terms inare nonnegative nearby. For $r<0$, the surface is locally a double cone with vertex at that point. Neither neighbourhood is homeomorphic to an open disc, so