Totally quasi-umbilic timelike surfaces in $\mathbb{R}^{1,2}$

TL;DR

This paper classifies all totally quasi-umbilic timelike surfaces in Minkowski space, revealing differences from Euclidean geometry where only planes and spheres are totally umbilic.

Contribution

It provides a complete classification of totally quasi-umbilic timelike surfaces in Minkowski space, expanding understanding of surface geometry in Lorentzian manifolds.

Findings

Identification of quasi-umbilic points where $H^2=K$ but the surface is not umbilic

Complete classification of totally quasi-umbilic timelike surfaces in $ ext{R}^{1,2}$

Demonstration of differences between Euclidean and Minkowski surface geometries

Abstract

For a regular surface in Euclidean space $\mathbb{R}^3$, umbilic points are precisely the points where the Gauss and mean curvatures $K$ and $H$ satisfy $H^2=K$; moreover, it is well-known that the only totally umbilic surfaces in $\mathbb{R}^3$ are planes and spheres. But for timelike surfaces in Minkowski space $\mathbb{R}^{1,2}$, it is possible to have $H^2=K$ at a non-umbilic point; we call such points {\em quasi-umbilic}, and we give a complete classification of totally quasi-umbilic timelike surfaces in $\mathbb{R}^{1,2}$.