Bifurcations of simple umbilical points defined by vector fields normal to a surface immersed in \(\mathbb{R}^4\) (Q1773061)

Let \(M\) be a two-dimensional surface in Euclidean space \(\mathbb{R}^4\). Then with respect to a fixed normal vector field \(\nu\) one can consider the shape operator \(S_\nu\), which defines the \(\nu\)-principal configuration on \(M\), that means the \(\nu\)-principal curvatures, \(\nu\)-principal lines and \(\nu\)-umbilical points. In many respects the situation is comparable with the surface theory in \(\mathbb{R}^3\). It is the purpose of this paper to investigate the \(\nu\)-principal configuration in the neighbourhood of an isolated \(\nu\)-umbilical point, in particular under a suitable variation of the normal field \(\nu\).