Positive quadratic differential forms: topological equivalence through Newton Polyhedra

A germe of a positive quadratic differential form in the plane can be written as$$\omega = a(x,y)dy^2+b(x,y)dxdy+c(x,y)dx^2,$$where $a$, $b$, $c \in C^{\infty}$. This kind of differential form appears in the study of the curvature lines and asymptotic curves in surfaces and in partial differential equations (in gas dynamics). Here, we show that this germ is topologically equivalent to its principal part, defined by the Newton Polyhedra, under suitable conditions.