On Lorentzian surfaces in $\mathbb{R}^{2,2}$
Abstract
We study the second order invariants of a Lorentzian surface in 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.
Cite
@article{arxiv.1503.06225,
title = {On Lorentzian surfaces in $\mathbb{R}^{2,2}$},
author = {Pierre Bayard and Victor Patty and Federico Sánchez-Bringas},
journal= {arXiv preprint arXiv:1503.06225},
year = {2015}
}