A Note on Umbilic Points at Infinity

TL;DR

This paper introduces a new, stronger definition of umbilic points at infinity for polynomial graphs in Euclidean 3-space, proving their properties and geometric interpretation, and establishing their existence in such surfaces.

Contribution

It proposes a novel definition of umbilic points at infinity, distinguishes their types, and proves their existence and properties for homogeneous polynomial graphs.

Findings

Umbilic points at infinity are isolated and occur in pairs.

Such points are zeros of the projective extension of the third fundamental form.

Homogeneous polynomial graphs necessarily have umbilic points at infinity.

Abstract

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 form, as developed by the authors in a previous paper. A geometric interpretation for our definition is that an umbilic point at infinity occurs when the tangent to the level set at infinity is also an asymptotic direction at infinity. We prove that a homogeneous polynomial graph must have an umbilic point, albeit at infinity.