The behavior of the principal distribution on a real-analytic surface
The author continues his earlier investigations concerning the behaviour of the index \(\text{ind}_0(G_F)\) of the curvature lines of the graph \(G_F\) of an analytic function \(z= F(x, y)\) in the neighbourhood of an isolated umbilical point \(o= (0,0,0)\) of \(G_F\) (\(F(0,0)= F_x(0,0)= F_y(0,0)= 0\), compare e.g. \textit{N. Ando} [The index of an isolated umbilical point on a surface, Contemp. Math. 308, 1--11 (2002; Zbl 1035.53005)]). His results culminate in an explicit formula for \(\text{ind}_0(G_F)\) for a graph \(G_F\), being an analytic and (in the sense of Hartman-Wintner) ``special'' Weingarten surface. This formula generalizes H. Hopf's classical result \(\text{ind}_0(G_F)< 0\) for analytical surfaces of constant mean curvature.
graph representation of an analytic surface
scientific article; zbMATH DE number 2092015