The behavior of the principal distributions on the graph of a homogeneous polynomial
type: Item
instance of: scholarly article
title: The behavior of the principal distributions on the graph of a homogeneous polynomial
zbMATH Open document ID: 1022.53006
author: Naoya Ando
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2003-10-26T00:00:00Z
review text: 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.
reviewed by: Anton Gfrerrer
Mathematics Subject Classification ID: 53A05
Mathematics Subject Classification ID: 57R99
zbMATH DE Number: 1827628
zbMATH Keywords: umbilical point
zbMATH Keywords: principal curvature distribution
zbMATH Keywords: homogeneous polynomial
zbMATH Keywords: index
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1113247561
OpenAlex ID: W2089983238
cites work: An isolated umbilical point of the graph of a homogeneous polynomial
cites work: The behavior of the principal distributions around an isolated umbilical point
cites work: Planar vector field versions of Carathéodory's and Loewner's conjectures
cites work: Umbilical Points and W-Surfaces
cites work: Differential geometry in the large. Seminar lectures New York University 1946 and Stanford University 1956. With a preface by S. S. Chern.
cites work: On G. Bol's proof of Carathéodory's conjecture
cites work: A Sharp Geometric Estimate for the Index of an Umbilic on a Smooth Surface
cites work: Real solvability of the equation \(\partial^2_{\overline z}\omega=\rho g\) and the topology of isolated umbilics
cites work: A proof of a conjecture of Loewner and of the conjecture of Caratheodory on umbilic points
DOI: 10.2748/TMJ/1113247561
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description: scientific article; zbMATH DE number 1827628