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Axiumbilic Singular Points on Surfaces Immersed in R4 and their Generic Bifurcations

Here are described the axiumbilic points that appear in generic one parameter families of surfaces immersed in R4. At these points the ellipse of curvature of the immersion, Little, Garcia - Sotomayor has equal axes. A review is made on the basic preliminaries on axial curvature lines and the associated axiumbilic points which are the singularities of the fields of principal, mean axial lines}, axial crossings and the quartic differential equation defining them. The Lie-Cartan vector field suspension of the quartic differential equation, giving a line field tangent to the Lie-Cartan surface (in the projective bundle of the source immersed surface which quadruply covers a punctured neighborhood of the axiumbilic point) whose integral curves project regularly on the lines of axial curvature. In an appropriate Monge chart the configurations of the generic axiumbilic points, denoted by E3, E4 and E5, are obtained by studying the integral curves of the Lie-Cartan vector field. Elementary bifurcation theory is applied to the study of the transition and elimination between the axiumbilic generic points. The two generic patterns E^1_{34} and E^1_{45} are analysed and their axial configurations are explained in terms of their qualitative changes (bifurcations) with one parameter in the space of immersions, focusing on their close analogy with the saddle-node bifurcation for vector fields in the plane . This work can be regarded as a partial extension to R4 of the umbilic bifurcations in Garcia - Gutierrez - Sotomayor for surfaces in R3. With less restrictive differentiability hypotheses and distinct methodology it has points of contact with the results of Gutierrez - Guinez - Castaneda.