2003/02/17 by Ronaldo Garcia, Garcia, Ronaldo, Jorge Sotomayor +1
Mathematics · Physics and Astronomy · #34D30 #37C75 #53A05 #53C12 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG #math.DS #msc:34D30 #msc:37C75 #msc:53A05 #msc:53C12
paper · pdf · doi:10.48550/arxiv.math/0302194
21 pages, 5 figures. To appear in Annales de la Faculte de Sciences de Toulouse
arxiv created 2003/02/17 · openalex publication_date 2003/02/17 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 R3. The leaves of the foliations are the lines of geometric mean curvature, along which the normal curvature is given by √ \mathcal K, which is the geometric mean curvature of the principal curvatures k1, k2 of the immersion. The singularities of the foliations are the umbilic points and parabolic curves, where k1 = k2 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.