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 ${ \mathbb R}^4$.
Categories: math.DG math.DS