Umbilic Singularities and Lines of Curvature on Ellipsoids of ${\mathbb R}^4$

TL;DR

This paper explores the topological structure of lines of principal curvature and umbilic singularities on three-dimensional ellipsoids in four-dimensional space, providing a comprehensive description of their geometric properties.

Contribution

It offers a detailed topological analysis of umbilic and partially umbilic singularities on 3D ellipsoids in 4D space, a novel extension in differential geometry.

Findings

Classification of umbilic singularities on ellipsoids in ${ m R}^4$

Description of lines of principal curvature topology

Identification of singularity types and their arrangements

Abstract

The topological structure of the lines of principal curvature, the umbilic and partially umbilic singularities of all tridimensional ellipsoids of ${\mathbb R}^4$ is described.