Lines of Curvature on Quadric Hypersurfaces of $ \mathbb{R}^4$

TL;DR

This paper investigates the geometric properties of lines of principal curvature and singularities on specific non-compact quadric hypersurfaces in four-dimensional space, extending previous work on compact ellipsoids.

Contribution

It provides a detailed analysis of curvature lines and umbilic singularities on non-compact quadric hypersurfaces in our-dimensional space, complementing prior studies on ellipsoids.

Findings

Characterization of principal curvature lines on hyperboloids

Identification of partially umbilic singularities

Extension of geometric analysis to non-compact quadrics

Abstract

Here are described the geometric structures of the lines of principal curvature and the partially umbilic singularities of the tridimensional non compact generic quadric hypersurfaces of ${\mathbb R}^4$. This includes the ellipsoidal hyperboloids of one and two sheets and the toroidal hyperboloids. The present study complements the analysis of the compact ellipsoidal hypersurfaces carried out by D. Lopes, R. Garcia and J. Sotomayor.