On Lorentzian surfaces in ℝ 2,2

We study the second-order invariants of a Lorentzian surface in ℝ 2,2 , and the curvature hyperbolas associated with 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...

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Published inProceedings of the Royal Society of Edinburgh. Section A. Mathematics Vol. 147; no. 1; pp. 61 - 88
Main Authors Bayard, Pierre, Patty, Victor, Sánchez-Bringas, Federico
Format Journal Article
LanguageEnglish
Published 01.02.2017
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Summary:We study the second-order invariants of a Lorentzian surface in ℝ 2,2 , and the curvature hyperbolas associated with 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.
ISSN:0308-2105
1473-7124
DOI:10.1017/S0308210516000147