The behavior of the principal distributions around an isolated umbilical point. (English) Zbl 0992.53006

Let \(G_f\) be the graph of a homogeneous polynomial \(f\) in two variables of degree \(k\geq 3\) such that the origin \(o\) of \(R^3\) is an isolated umbilical point. If \(\Theta_0\) is a real number at which \({df\over d\Theta} (\cos\Theta,\sin\Theta)\) vanishes, the straight line \(L(\Theta_0): =\{(x,y)\in R^2\); \(x\sin\Theta_0 -y\cos\Theta_0 =0\}\) is called a root line of \(f\). The main purpose of the paper is to derive the existence of further umbilics of \(G_f\) on \(L(\Theta_0) \setminus \{o\}\) from assumptions concerning the curvature line distribution around \(o\) and the behaviour of the gradient vector field of \(f\).


53A05 Surfaces in Euclidean and related spaces
53B25 Local submanifolds
53A99 Classical differential geometry
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