Encontros IST-IME 

09/09/2016, 11:20 — 12:00 — Anfiteatro Pa2, Pavilhão de Matemática
Ronaldo Garcia, IME, Universidade Federal de Goiás

Partially Umbilic Singularities of Hypersurfaces of $\mathbb{R}^4$ and of Plane Fields of $\mathbb{R}^3$

In this talk will be established 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 at 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 $D_{23}$.

Also a similar result is obtained for principal lines associated to plane fields of $\mathbb{R}^3$.