  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.
