ORCID
https://orcid.org/0000-0002-7437-5046
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
Disciplines
Algebra | Geometry and Topology | Mathematics | Physical Sciences and Mathematics
DOI
10.1007/s13366-024-00740-3
Publisher
Springer
License Condition
This work is licensed under a Creative Commons Attribution 4.0 International License.
Recommended Citation
Guilfoyle, B. A note on umbilic points at infinity. Beitr Algebra Geom (2024). https://doi.org/10.1007/s13366-024-00740-3
Publication Details
Contributions to Algebra and Geometry, 2024. © The Author(s) 2024.