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On Lorentzian surfaces in ℝ2,2

Published online by Cambridge University Press:  06 January 2017

Pierre Bayard
Affiliation:
Facultad de Ciencias, Universidad Nacional Autónoma de México, México ([email protected])
Victor Patty
Affiliation:
Instituto de Física y Matemáticas, UMSNH, Ciudad Universitaria, CP 58040 Morelia, Michoacán, México ([email protected])
Federico Sánchez-Bringas
Affiliation:
Facultad de Ciencias, Universidad Nacional Autónoma de México, México ([email protected])

Extract

We study the second-order invariants of a Lorentzian surface in ℝ2,2, 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.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2017 

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