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Dupin indicatrices and families of curve congruences.

Bruce, J. W. and Tari, F. (2005) 'Dupin indicatrices and families of curve congruences.', Transactions of the American Mathematical Society., 357 (1). pp. 267-285.


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

Item Type:Article
Keywords:Implicit differential equations, Differential geometry.
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Publisher statement:First published in Transactions of the American Mathematical Society 357(1) 2005, published by the American Mathematical Society
Record Created:27 Aug 2008
Last Modified:24 Aug 2011 13:25

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