Rank-1 codimension one singularities of positive quadratic differential forms

Abstract

We complete the study of first-order structural stability at singular points of positive quadratic differencial forms on two manifolds. For this, we consider the generic 1-parameter bifurcation of a D 23 -singular point. This situation consists in having, before the bifurcation, two locally stable singular points (one of type D 2 and the other of type D 3 ) which collapse at the D 23 -singular point when the bifurcation parameter is reached, and afterwards disappear. In local ( x , y ) -coordinates, such a point appears at the origin of a planar differential equation of the form a ( x , y ) dy 2 + 2 b ( x , y ) dx dy + c ( x , y ) dx 2 , with ( b 2 - ac ) ( x , y ) ⩾ 0 , such that (1) the first jet of the map ( a , b , c ) at the origin is T 1 ( a , b , c ) ( 0 , 0 ) = ( y , 0 ,- y ) and (2) ∂ 2 b ∂ x 2 ≠ 0 .