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Corpus ID: 237267300

A Note on Umbilic Points at Infinity

@inproceedings{Guilfoyle2021ANO,
title={A Note on Umbilic Points at Infinity},
author={Brendan Guilfoyle and Adriana Ortiz-Rodr{\'i}guez},
year={2021}
}

In this note a definition of umbilic point at infinity is proposed, at least for surfaces that are homogeneous polynomial graphs over a plane in Euclidean 3-space. This is a stronger definition than that of Toponogov in his study of complete convex surfaces, and allows one to distinguish between different umbilic points at infinity. It is proven that all such umbilic points at infinity are isolated, that they occur in pairs and are the zeroes of the projective extension of the third fundamental… Expand

The global qualitative behaviour of fields of principal directions for the graph of a real valued polynomial function $f$ on the plane are studied. We provide a Poincar\'e-Hopf type formula where the… Expand

Counter-examples to the famous conjecture of Caratheodory, as well as the bound on umbilic index proposed by Hamburger, are constructed with respect to Riemannian metrics that are arbitrarily close… Expand

In the 1950's Hopf gave examples of non-round convex 2-spheres in Euclidean 3-space with rotational symmetry that satisfy a linear relationship between their principal curvatures. In this paper we… Expand

We survey existence and classication of totally umbilic surfaces in the model geometries of Thurston and the Berger spheres. We also discuss the regularity of totally umbilic surfaces.

Spivak's Comprehensive introduction takes as its theme the classical roots of contemporary differential geometry. Spivak explains his Main Premise (my term) as follows: "in order for an introduction… Expand