Versal Unfoldings for rank--2 singularities of positive quadratic differential forms: The remaining case

Abstract

We complete the local study of rank--2 singular points of positive quadratic differential forms on oriented two--dimensional manifolds. We associate to each positive quadratic differential form $\omega$ defined on an oriented two--dimensional manifold $M$ two transversal one--dimensional foliations $f_1(\omega)$ and $f_2(\omega)$ with common set of singular points. This study was begun in [Gut-Gui] for a generic class of singularities called <em>simple</em>, and continued in [Gui-Sa] for those non--simple rank--2 singular points called of <em>type C</em>. Taking into account the classification of [Gui3], the only rank--2 singular points which remain to be studied are those of type E($\lambda$), for $\lambda\geq 1 $. We undertake the local study of the remaining case under a non--flatness condition on the positive quadratic differential form at the singular point.