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

Ando, Naoya

J. Math. Soc. Japan 53, No. 1, 237-260 (2001).

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 \(f\).

Reviewer: K.Leichtweiß (Stuttgart)

MSC:

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

Keywords:

umbilics; root lines; signs and critical signs of roots; gradient roots