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

DOI:
https://doi.org/10.1090/S0002-9947-04-03497-X

Published electronically:
April 16, 2004

MathSciNet review:
2098095

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study a number of natural families of binary differential equations (BDE's) on a smooth surface in . 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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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

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:
https://doi.org/10.1090/S0002-9947-04-03497-X

Keywords:
Implicit differential equations,
differential geometry

Received by editor(s):
February 4, 2003

Received by editor(s) in revised form:
July 23, 2003

Published electronically:
April 16, 2004

Article copyright:
© Copyright 2004
American Mathematical Society