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Erschienen in:
24.10.2015
On the Singularities of Families of Curve Congruences on Lorentzian Surfaces
Erschienen in: Journal of Dynamical and Control Systems | Ausgabe 3/2016
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Abstract
In Nabarro (2011), we define and study the families of 𝔸-conjugate curve congruences \({\mathcal {L}\mathcal {C}}^{i}_{\alpha }\) and the families of reflected 𝔸-conjugate curve congruences \({\mathcal {L}\mathcal {R}}^{i}_{\alpha }\), i = 1, 2, associated to a self-adjoint operator 𝔸 on a smooth and oriented surface M endowed with a Lorentzian metric. These families parametrize parts of the pencils of forms that link the equation of the 𝔸-asymptotic (resp. 𝔸-characteristic) curves and that of the 𝔸-principal curves. There is a crucial difference with the Riemannian case due to the existence of lightlike curves. In this paper, we study the generic local singularities in the members of these families and describe the way they bifurcate within the families.