Positive Differential 2-Forms

Abstract

Let U ⊑ M be a local chart on M and let ω(u, v) be a C r -quadratic differential form: \(a(u,v)d{u^2} + b(u,v)dudv + c(u,v)d{v^2},\) where a, b and c are real-valued functions of class C r . By a positive C r differential 2-form on M one understands a C r -quadratic differential form ω such that for every point x ∈ M, the set ω -1(x)(0) is either A union of two transversal lines (such a point is called regular),or An isolated point (such a point is called singular).