2014/10/30 by Débora Lopes, Lopes, Débora, Jorge Sotomayor +3
Mathematics · #37C15 #53C12 #57R30 #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Holomorphic and Operator Theory #math.DG #math.DS #msc:37C15 #msc:53C12 #msc:57R30
paper · pdf · doi:10.48550/arxiv.1410.8548
50 pages and 17 figures
arxiv created 2014/10/30 · openalex publication_date 2014/10/30 · arxiv updated 2014/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper establishes the geometric structure of the lines of principal curvature of a hypersurface immersed in \mathbb R4 in a neighborhood of the set S of its principal curvature singularities, consisting of the points at which atF least two principal curvatures are equal. Under generic conditions defined by appropriate transversality hypotheses it is proved that S is the union of regular smooth curves S12 and S23, consisting of partially umbilic points, where only two principal curvatures coincide. This curve is partitioned into regular arcs consisting of points of Darbouxian types D1, D2, D3, with common boundary at isolated semi-Darbouxian transition points of types D12 and D23. The stratified structure of the partially umbilic separatrix surfaces, consisting of the boundary of the set of points through which the principal lines approach \mathcal S, established in this work, extends to hypersurfaces in \mathbb R4 the results of Darboux for umbilic points on analytic surfaces in \mathbb R3, reformulated by Gutierrez and Sotomayor, to describe the umbilic separatrix structures of the umbilic types D1, D2, D3, and further developed by Garcia, Gutierrez and Sotomayor, for their D12 and D23 generic bifurcations. This work complements results of Garcia on the structure of principal curvature lines around the generic partially umbilic points of hypersurfaces in \mathbb R4.