The behavior of the principal distributions on the graph of a homogeneous polynomial
Let \(f = f(x,y)\) be a homogeneous polynomial of degree \(k\) greater than 2 and let \(G_f := \{(x,y,f(x,y))\}\) denote its graph in \({\mathbb R}^3\). Let moreover \(\text{Umb}(G_f)\) denote the set of umbilical points of \(G_f\) -- i.e. the points of \(G_f\) in which all normal curvatures are equal. In any point \(p\) in \(G_f\backslash \text{Umb}(G_f)\) there exist exactly two directions of extremal curvature. Their distribution in \(G_f\backslash \text{Umb}(G_f)\) is described by the two principal distributions \(D_f^{(1)}\), \(D_f^{(2)}\). In the paper the author studies the behavior of \(D_f^{(1)}\), \(D_f^{(2)}\) in the neighbourhood of an umbilical point \(p\) by means of its so-called index \(\text{ind}_{p_0}(G_f)\) as defined in [\textit{H. Hopf}, Lectures on differential geometry in the large, Lect. Notes Math. 1000, Springer Verlag, Berlin (1989; Zbl 0669.53001)]. He gives the possible values of the index of any umbilical point and after one-point compactification also those of the point at infinity.
scientific article; zbMATH DE number 1827628