Let $F:R^3->R$ given by $F(x,y,z)=ax^2+by^2+cz^2 - 1$
I wanto to proof that inverse image of F apply at point $0$ is a regular surface with curvature mean constant.
Let me tell you what my first steps towards solution. If I'm making a mistake, let me know
Resolution steps:
1) First of all, we need calcule the Hessian Matrix $H$ of $F$
2) $H$ is symetric, so we would like to calcule the Eigenvalues and Eigenvectors of $H$.
3) The eigenvalues of H are called principal curvatures and are invariant under rotation. They are denoted $κa$ and $κb$.
I need confess, the questions is not very clear, to me is more confortable to know what coeficients i'm working, because this way, i knew what kind of quadratic surface is. Perhaps we should discuss what conditions $a, b, c$ will have to regular surface with CMC