The behavior of the principal distributions around an isolated umbilical point

Abstract

Let f be a homogeneous polynomial in two variables such that on its graph G_f, the origin o=(0, 0, 0) of \bm{R}³ is an isolated umbilical point. In this paper, the behavior of the principal distributions around o is studied in relation to the existence of other umbilical points than o and the behavior of the gradient vector field of f.