a1 Department of Mathematics, University of Newcastle upon Tyne, Newcastle NE1 7RU, U.K.
In this paper we give the local classification of solution curves of bivalued direction fields determined by the equation
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)