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 $0=(0,0,0)$ of $R^3$ is an isolated umbilical point. In this paper, the behavior of the principal distributions around $0$ is studied in relation to the existence of other umbilical points than $0$ and the behavior of the gradient vector field of $f$.