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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Dupin indicatrices and families of curve congruences

Authors: J. W. Bruce and F. Tari
Journal: Trans. Amer. Math. Soc. 357 (2005), 267-285
MSC (2000): Primary 53A05, 34A09
Published electronically: April 16, 2004
MathSciNet review: 2098095
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Abstract: We study a number of natural families of binary differential equations (BDE's) on a smooth surface $M$ 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 $M$, 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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Additional Information

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

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

Keywords: Implicit differential equations, differential geometry
Received by editor(s): February 4, 2003
Received by editor(s) in revised form: July 23, 2003
Published electronically: April 16, 2004
Article copyright: © Copyright 2004 American Mathematical Society