Lagrange mappings of the first open Whitney umbrella

In this paper we give a classification of simple stable singularities of Lagrange projections of the first open Whitney umbrella, the simplest singularity of Lagrange varieties. Our classification extends the ADE-classification, due to Arnold, of simple stable singularities of Lagrange projections of smooth Lagrange submanifolds. We also prove a criterion of equivalence of stable Lagrange projections of open Whitney umbrellas which is analogous to Mather's fundamental theorem on stable map-germs.