Structurally stable configurations of lines of mean curvature and umbilic points on surfaces immersed in \(\mathbb{R}^3\)
The present paper concerns the mean curvature configurations associated to immersions of an oriented surface into \(\mathbb{R}^3\), i.e. pairs of orthogonal foliations whose singularities and leaves are given respectively by the umbilic points and the lines of normal mean curvature of the immersion. These lines are characterized by the condition that along them the immersions bend according to their normal mean curvature. Using the formal analogy between mean and principal configurations, the authors study the stable and generic cases of umbilic points and mean curvature cycles, providing in this way sufficient conditions for mean curvature structural stability. Moreover, examples of mean curvature configurations on the ellipsoid and torus are given.
scientific article; zbMATH DE number 1740176