Lines of curvature of algebraic surfaces
Let \(S\subset{\mathbb R}^3\) be a real algebraic surface. At each smooth point of \(S\), where the two principal curvatures are different, consider the two orthogonal principal curvature directions. The corresponding integral curves on \(S\) are the lines of minimal and maximal principal curvature. The surface \(S\) is called structurally stable, if for every small perturbation \(S'\) of \(S\), there is a homeomorphism between \(S\) and \(S'\), which maps the lines of minimal and maximal principal curvature of \(S\) onto those of \(S'\). The authors prove that a generic surface of degree \(d\geq3\) is structurally stable.
scientific article; zbMATH DE number 1013720