On Lorentzian surfaces in $\mathbb{R}^{2,2}$

TL;DR

This paper investigates the second order invariants, curvature hyperbolas, and Gauss map of Lorentzian surfaces in ^{2,2}, revealing new invariants, properties of asymptotic directions, and special classes like quasi-umbilic surfaces.

Contribution

It introduces new invariants for Lorentzian surfaces, analyzes their geometric properties, and characterizes special surface classes such as quasi-umbilic surfaces.

Findings

Identification of new invariants in degenerate cases

Relation between asymptotic directions and causal character

Characterization of surfaces with vanishing classical invariants

Abstract

We study the second order invariants of a Lorentzian surface in $\mathbb{R}^{2,2},$ and the curvature hyperbolas associated to its second fundamental form. Besides the four natural invariants, new invariants appear in some degenerate situations. We then introduce the Gauss map of a Lorentzian surface and give an extrinsic proof of the vanishing of the total Gauss and normal curvatures of a compact Lorentzian surface. The Gauss map and the second order invariants are then used to study the asymptotic directions of a Lorentzian surface and discuss their causal character. We also consider the relation of the asymptotic lines with the mean directionally curved lines. We finally introduce and describe the quasi-umbilic surfaces, and the surfaces whose four classical invariants vanish identically.