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October  2005, 12(5): 887-904. doi: 10.3934/dcds.2005.12.887

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

Víctor Guíñez 1, and  Eduardo Sáez 2,

1. 

Universidad de Santiago de Chile, Departamento de Matemática y C.C., Casilla 307, Correo 2, Santiago
2. 

Universidad Técnica Federico Santa María, Departamento de Matemática, Casilla 110-V, Valparaíso, Chile

Received  January 2004 Revised  November 2004 Published  February 2005

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 simple, and continued in [Gui-Sa] for those non--simple rank--2 singular points called of type C. 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.
Keywords: singular points, foliations, Quadratic differential forms, versal unfoldings.
Mathematics Subject Classification: 57R30, 34C20, 37G1.
Citation: Víctor Guíñez, Eduardo Sáez. Versal Unfoldings for rank--2 singularities of positive quadratic differential forms: The remaining case. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 887-904. doi: 10.3934/dcds.2005.12.887
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