A Note on Umbilic Points at Infinity

Authors

Brendan Guilfoyle, School of STEM, Munster Technological University, Kerry, Tralee Co., Kerry, IrelandFollow

ORCID

https://orcid.org/0000-0002-7437-5046

Document Type

Article

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Disciplines

Algebra | Algebraic Geometry | Mathematics | Physical Sciences and Mathematics

Publication Details

Beiträge zur Algebra und Geometrie. © The Author(s) 2024

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 in Guilfoyle and Ortiz-Rodríguez (Math Proc R Ir Acad 123A(2), 63–94, 2023). A geometric interpretation for the 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.

Recommended Citation

Guilfoyle, B. A note on umbilic points at infinity. Beitr Algebra Geom (2024). https://doi.org/10.1007/s13366-024-00740-3