Hostname: page-component-7b9c58cd5d-7g5wt Total loading time: 0 Render date: 2025-03-15T14:58:30.384Z Has data issue: false hasContentIssue false

Families of curve congruences on Lorentzian surfaces and pencils of quadratic forms

Published online by Cambridge University Press:  03 June 2011

Ana Claudia Nabarro
Affiliation:
ICMC-USP, Dept. de Matemática, Avenida do Trabalhador São-Carlense, 400 Centro, Caixa Postal 668, CEP 13560-970, São Carlos (SP), Brazil (anaclana@icmc.usp.br)
Farid Tari
Affiliation:
Department of Mathematical Sciences, Durham University, Science Laboratories, South Road, Durham DH1 3LE, UK (farid.tari@durham.ac.uk)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We define and study families of conjugate and reflected curve congruences associated to a self-adjoint operator on a smooth and oriented surface M endowed with a Lorentzian metric g. These families trace parts of the pencil joining the equations of the -asymptotic and the -principal curves, and the pencil joining the -characteristic and the -principal curves, respectively. The binary differential equations (BDEs) of these curves can be viewed as points in the projective plane. Using the polar lines of various BDEs with respect to the conic of degenerate quadratic forms, we obtain geometric results on the above pencils and their relation with the metric g, on the type of solutions of a given BDE, of its -conjugate equation and on BDEs with orthogonal roots.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2011