The umbilical locus on the boundary of strictly pseudoconvex domains in ℂ2

The main objective of this paper is to survey some recent results on the Chern–Moser question concerning existence of umbilical points on three-dimensional CR submanifolds in [Formula: see text].

Hypersurfaces with Two Distinct Para-Blaschke Eigenvalues inSn+1(1)

Letx:M↦Sn+1(1)be ann (n≥3)-dimensional immersed hypersurface without umbilical points and with vanishing Möbius form in a unit sphereSn+1(1), and letAandBbe the Blaschke tensor and the Möbius second fundamental form ofx, respectively. We define a symmetric(0,2)tensorD=A+λBwhich is called the para-Blaschke tensor ofx, whereλis a constant. An eigenvalue of the para-Blaschke tensor is calleda para-Blaschke eigenvalueofx. The aim of this paper is to classify the oriented hypersurfaces inSn+1(1)with two distinct para-Blaschke eigenvalues under some rigidity conditions.

Lines of Curvature and Umbilical Points: Computation Method and Applications

A method for computinglines of curvatureandumbilical pointsis proposed. These properties, derived for NURBS surfaces, are useful in shape modeling for both aesthetic and functional characteristics evaluation. Moreover, the application to theship-hull designand to theprogressive additional lens design, of umbilics and lines of curvature are investigated.