Dynamics of principal configurations near umbilics for surfaces in $mathbb(R)^4$

Abstract

The $v$-principal configuration of an immersed surface $M$ in $\mathbb(R)^4$ is the set formed by the umbilical points and the lines of principal curvatures with respect to an unitary smooth vector field $v$ normal to $M$. In this article we describe the bifurcation set of $v$-principal configurations of a local surface $M$ depending on two parameters of the surface and depending also on the 1-jet of the vector field $v$ normal to $M$ which defines an isolated simple umbilical point of $M$.