Parabolic, ridge and sub-parabolic curves on implicit surfaces with singularities (Q1687144)

A function germ \(f:(\mathbb R^n,0)\to(\mathbb R,0)\) has a Morse singularity at \(0\) if its first partial derivatives vanish at \(0\) and the determinant of the Hessian matrix \(\mathcal H_f\) does not vanish at \(0\). The index of the Morse singularity \(0\) is the number of the negative eigenvalues of \(\mathcal H_f(0)\). In this paper, by presenting local parameterizations of the surfaces, the author studies parabolic, ridge and sub-parabolic curves on implicit surfaces defined by functions having Morse singularities of index \(1\), and shows the asymptotic behavior of the principal curvatures and directions. In the appendix, he also presents height and distance squared functions on implicit surfaces.