Geometric Mean Curvature Lines on Surfaces Immersed in R3

Abstract

Here are studied pairs of transversal foliations with singularities, defined on the Elliptic region (where the Gaussian curvature $\mathcal K$ is positive) of an oriented surface immersed in $\mathbb R^3$. The leaves of the foliations are the lines of geometric mean curvature, along which the normal curvature is given by $\sqrt {\mathcal K}$, which is the geometric mean curvature of the principal curvatures $ k_1, k_2$ of the immersion. The singularities of the foliations are the umbilic points and parabolic curves}, where $ k_1 = k_2$ and ${\mathcal K} = 0$, respectively. Here are determined the structurally stable patterns of geometric mean curvature lines near the umbilic points, parabolic curves and geometric mean curvature cycles, the periodic leaves of the foliations. The genericity of these patterns is established. This provides the three essential local ingredients to establish sufficient conditions, likely to be also necessary, for Geometric Mean Curvature Structural Stability. This study, outlined at the end of the paper, is a natural analog and complement for the Arithmetic Mean Curvature and Asymptotic Structural Stability of immersed surfaces studied previously by the authors.