There are a number of classical binary differential equations (BDE's) associated to surfaces in 3-space. These include those determined by the asymptotic directions, the principal directions and, perhaps less well known, the characteristic directions, a natural analogue of the asymptotic directions which live on the elliptic part of the surface. Of special interest are those points where the BDE's have zeros. For the asymptotic directions these are the cusps of Gauss; for the principal directions they are the umbilics; for the characteristic directions they are both. Now there are natural involutions on the tangent space to a surface in 3-space which in turn provide natural families of BDE's linking asymptotic and principle (resp. characteristic and principle) directions. These families have zeros and the resulting zero curves (joining for example cusps of Gauss with umbilics) are of some interest. Surprisingly the zero curve corresponding to the family joining the asymptotic and principal directions (resp. characteristic and principle directions) are naturally linked to the characteristic (resp. asymptotic) directions. In other words these three classical BDE's are intimately connected.
There is an explanation for this relationship which holds for a new class, the so-called curvature BDE's, and involves some classical facts concerning the space of quadratic binary forms (with coefficients in the field of rational functions in the principal curvatures) and self-polar triangles.
The talk only assumes a basic knowledge of the differential geometry of surfaces in 3-space.
This is joint work with Farid Tari (University of Durham) and will appear in the Transactions of the AMS.
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Last updated April 29, 2004. This page was created and is maintained by Stephen Wills (s.wills@ucc.ie).