Lines of axial curvature on surfaces immersed in R4

AbstractThe main notions concerning umbilics and lines of principal curvature, traditionally studied on surfaces in R3 , are extended in this paper to surfaces immersed in R4 . Departing from the classical second fundamental form and the ellipse of curvature of the immersion of a surface into R4 here are studied in detail the concepts of a) axiumbilic points, analogous to classical umbilics since at them the ellipse of curvature has equal axes, and b) periodic lines of axial curvature, called here axial cycles, corresponding both to principal and mean curvature cycles in the classical R3 case. To any immersed surface in R4 its axial configurations: the principal configuration and the mean curvature configuration are associated. For surfaces in R3 , the first one reduces to the configuration by umbilics and principal lines, while the second one gives the configuration by umbilics and integral foliations of the mean curvature line fields.Also the notion of principal structural stability of immersions of surfaces into R3 is extended to that of axial structural stability, for the case of surfaces in R4 . Sufficient conditions for the axial structural stability are provided in terms of axiumbilics, axial cycles and the asymptotic behavior of all the other lines of axial curvatur