Transactions of the American Mathematical Society

Author(s): J. W. Bruce; F. Tari
Journal: Trans. Amer. Math. Soc. 357 (2005), 267-285.
MSC (2000): Primary 53A05, 34A09
Posted: April 16, 2004
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Abstract: We study a number of natural families of binary differential equations (BDE's) on a smooth surface

in ${\mathbb{R}}^3$ . One, introduced by G. J. Fletcher in 1996, interpolates between the asymptotic and principal BDE's, another between the characteristic and principal BDE's. The locus of singular points of the members of these families determine curves on the surface. In these two cases they are the tangency points of the discriminant sets (given by a fixed ratio of principle curvatures) with the characteristic (resp. asymptotic) BDE.

More generally, we consider a natural class of BDE's on such a surface

, and show how the pencil of BDE's joining certain pairs are related to a third BDE of the given class, the so-called polar BDE. This explains, in particular, why the principal, asymptotic and characteristic BDE's are intimately related.

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Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53A05, 34A09

J. W. Bruce
Affiliation: Division of Pure Mathematics, Department of Mathematical Sciences, University of Liverpool, Mathematics and Oceanography Building, Peach Street, Liverpool L69 7ZL, United Kingdom
Address at time of publication: Deputy Vice-Chancellor, University of Hull, Cottingham Road, Hull HU6 7RX, United Kingdom
Email: jwbruce@liv.ac.uk, j.w.bruce@hull.ac.uk

F. Tari
Affiliation: Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Avenida Trabalhador Sãocarlense, 400 Centro, Caixa Postal 668, CEP 13560-970, São Carlos (SP), Brazil
Email: tari@icmc.usp.br

DOI: 10.1090/S0002-9947-04-03497-X
PII: S 0002-9947(04)03497-X
Keywords: Implicit differential equations, differential geometry
Received by editor(s): February 4, 2003
Received by editor(s) in revised form: July 23, 2003
Posted: April 16, 2004
Copyright of article: Copyright 2004, American Mathematical Society

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