Conformal invariants and spherical contacts of surfaces in ℝ4

Abstract

Based on the fact that conformal maps preserve contacts of surfaces with hyperspheres, we introduce the concept of strong principal lines on surfaces in R 4 and obtain conformally invariant differential 1-forms along them. The zeros of these 1-forms are respectively characterized as ridges (singularities of squared-distance functions of type Ak ,k ≥ 4) and higher order semiumbilics (singularities of type Dk ,k ≥ 5). As a consequence we obtain that any closed orientable surface generically immersed in R 4 has at least 2 semiumbilic points of type D5. We provide geometri- cal interpretations of these conformally invariant 1-forms in terms of the geometry of curves induced in the 5-dimensional de Sitter space and in the 5-dimensional light- cone.