Positive quadratic differential forms: Linearization, finite determinacy and versal unfolding
A quadratic differential form \[ \omega= a(x,y)dx^2+ 2b(x,y)dxdy +c(x,y)dy^2 \] on an oriented and connected two-dimensional manifold \(M\) is called positive if for every \(p\in M\), the subset \(\omega (p)^{-1} (0)\subset T_p(M)\) is either the union of two lines (then \(p\) is called a regular point) or the whole tangent space (singular point \(p)\). The authors study the equivalence, structural stability, linearization, finite determinacy, and versal unfolding around the so-called simple singular points of a positive form defined by the property that \[ a=y+ M_1(x,y), \quad b=b_1x+b_2y +M_2(x,y),\quad c= -y+M_3(x,y) \] \((b_1\neq 0\), \(M_i=O \sqrt {(x^2+y^2)}\), \(i=1,2,3)\) after appropriate linear transformation (the coordinate-free definition is possible). The simple points are generic and persistent.
scientific article; zbMATH DE number 1051570
scientific article