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arXiv · 1503.06225

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

Pierre Bayard·Victor Patty·Federico Sánchez-Bringas
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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.

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Pierre Bayard, Victor Patty, Federico Sánchez-Bringas. 2015-03-20. On Lorentzian surfaces in $\mathbb{R}^{2,2}$. https://arxiv.org/abs/1503.06225

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