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.
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