Partially umbilic singularities of hypersurfaces of $\mathbb R^4$

TL;DR

This paper analyzes the geometric structure of principal curvature lines near partially umbilic singularities on hypersurfaces in four-dimensional space, extending classical results and describing stratified structures and bifurcations.

Contribution

It establishes the stratified structure of partially umbilic points and separatrix surfaces on hypersurfaces in R^4, extending classical surface results to higher dimensions.

Findings

Partially umbilic set forms smooth curves with Darbouxian types.

Stratified structure of separatrix surfaces is characterized.

Results extend classical surface theory to hypersurfaces in R^4.

Abstract

This paper establishes the geometric structure of the lines of principal curvature of a hypersurface immersed in ${\mathbb R}^4$ in a neighborhood of the set $\mathcal{S}$ 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 $\mathcal{S}$ is the union of regular smooth curves $\mathcal{S}_{12}$ and $\mathcal{S}_{23}$, consisting of partially umbilic points, where only two principal curvatures coincide. This curve is partitioned into regular arcs consisting of points of Darbouxian types $D_1,\; D_2,\; D_3$, with common boundary at isolated semi-Darbouxian transition points of types $ D_{12}$ and $D_{23}$. 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 ${\mathbb R}^4$ the results of Darboux for umbilic points on analytic surfaces in ${\mathbb R}^3$, reformulated by Gutierrez and Sotomayor, to describe the umbilic separatrix structures of the umbilic types $D_1,\; D_2,\; D_3$, and further developed by Garcia, Gutierrez and Sotomayor, for their $ D_{12}$ and $D_{23}$ generic bifurcations. This work complements results of Garcia on the structure of principal curvature lines around the generic partially umbilic points of hypersurfaces in ${\mathbb R}^4$.