Versal unfolding for rank-2 singularities of positive quadratic differential forms: The remaining case
Authors study rank-2 singular points of positive quadratic differential forms (in local coordinates) \[ \omega = a(x,y)dy^2+2b(x,y)dxdy+c(x,y)dy^2 \] on oriented two-dimensional manifolds. To each positive quadratic differential form \(\omega \) defined on an oriented two-dimensional manifold they associate two transversal one-dimensional foliations with common set of singular points. The three different locally stable phase portraits and their normal forms, associated to these foliations around the singularity are obtained and the versal unfoldings of the normal forms are constructed and the respective codimensions are calculated. Using the blowing up techniques the case of rank-2 singular points with the parametrized form of singularity denoted by \(E(\lambda )\) is investigated in relatively complete form with finding a normal form of any family of positive quadratic differential forms by analytic change of coordinates and singularity type \(E(\lambda )\) at zero of parameter value.
scientific article; zbMATH DE number 2177528