Dupin indicatrices and families of curve congruences
Authors:
J. W. Bruce and F. Tari
Journal:
Trans. Amer. Math. Soc. 357 (2005), 267285
MSC (2000):
Primary 53A05, 34A09
Published electronically:
April 16, 2004
MathSciNet review:
2098095
Fulltext PDF Free Access
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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 socalled polar BDE. This explains, in particular, why the principal, asymptotic and characteristic BDE's are intimately related.
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 J.W. Bruce and D. Fidal, On binary differential equations and umbilics, Proc. Royal Soc. Edinburgh, 111A (1989), 147168. MR 90e:58141
 2.
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 3.
 J.W. Bruce, G. J. Fletcher and F. Tari, Zero curves of families of curve congruences, to appear in the Proceedings of the Workshop on Real and Complex Singularities (T. Gaffney & M.A.S. Ruas, Editors), ICMC  USP  São Carlos, Brazil, 29 July  02 August, 2002.
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 J.W. Bruce and F. Tari, Generic 1parameter families of binary differential equations of Morse type, Discrete and Continuous Dynamical Systems, Vol. 3, No. 1 (1997), 7990. MR 98h:58123
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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 ViceChancellor, 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 13560970, São Carlos (SP), Brazil
Email:
tari@icmc.usp.br
DOI:
http://dx.doi.org/10.1090/S000299470403497X
PII:
S 00029947(04)03497X
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
