Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

On binary differential equations and umbilics

J.W. Brucea1 and D.L. Fidala1

a1 Department of Mathematics, University of Newcastle upon Tyne, Newcastle NE1 7RU, U.K.

Synopsis

In this paper we give the local classification of solution curves of bivalued direction fields determined by the equation

S0308210500025087_eqnU1

where a and b are smooth functions which we suppose vanish at 0 ∈ ℝ2. Such fields arise on surfaces in Euclidean space, near umbilics, as the principal direction fields, and also in applications of singularity theory to the structure of flow fields and monochromatic-electromagnetic radiation. We give a classification up to homeomorphism (there are three types) but the methods furnish much additional information concerning the fields, via a crucial blowing-up construction.

(Received December 10 1987)

(Revised June 27 1988)