Lines of axial curvature at critical points on surfaces mapped into R4

TL;DR

This paper analyzes the behavior of axial curvature lines near critical points on surfaces mapped into four-dimensional space, describing how axial umbilic points move and bifurcate as parameters vary.

Contribution

It provides a detailed description of axial curvature line patterns and umbilic point bifurcations near critical points in mappings from surfaces to R4.

Findings

Characterization of axial curvature lines near critical points

Description of axial umbilic point movements and bifurcations

Full description of arc structures and umbilic types in typical mappings

Abstract

In this paper are studied the simplest patterns of axial curvature lines (along which the normal curvature vector is at a vertex of the ellipse of curvature) near a critical point of a surface mapped into R4. These critical points, where the rank of the mapping drops from 2 to 1, occur isolated in generic one parameter families of mappings of surfaces into R4. As the parameter crosses a critical bifurcation value, at which the mapping has a critical point, it is described how the axial umbilic points, which are the singularities of the axial curvature configurations at regular points, move along smooth arcs to reach the critical point. The numbers of such arcs and their axial umbilic types are fully described for a typical family of mappings with a critical point.