Partially umbilic singularities of hypersurfaces of R4R4

This paper establishes the geometric structure of the lines of principal curvature of a hypersurface immersed in R4R4 in a neighborhood of the set SS of its principal curvature singularities  , consisting of the points at which at least two principal curvatures are equal. Under generic conditions defined by appropriate transversality hypotheses it is proved that SS is the union of regular smooth curves S12S12 and S23S23, consisting of partially umbilic points  , where only two principal curvatures coincide. This curve is partitioned into regular arcs consisting of points of Darbouxian types D1D1, D2D2, D3D3, with common boundary at isolated semi-Darbouxian transition points of types D12D12 and D23D23. The stratified structure of the partially umbilic separatrix surfaces  , consisting of the boundary of the set of points through which the principal lines approach SS, established in this work, extends to hypersurfaces in R4R4 the results of Darboux in [1] for umbilic points on analytic surfaces in R3R3, reformulated by Gutierrez and Sotomayor in [8], to describe the umbilic separatrix structures of the umbilic types D1D1, D2D2, D3D3, and further developed by Garcia, Gutierrez and Sotomayor in [6], for their D12D12 and D23D23 generic bifurcations. This work complements results of Garcia [5] on the structure of principal curvature lines around the generic partially umbilic points of hypersurfaces in R4R4.