The structure of the set of singular points of a codimension $1$ differential system on a $5$-manifold

Abstract:

Generic modules $V$ of vector fields tangent to a 5-dimensional smooth manifold $M$, generated locally by four not necessarily linearly independent fields ${X_1}$, ${X_2}$, ${X_3}$, ${X_4}$, are considered. Denoting by $\omega$ the 1-form ${X_4}\lrcorner {X_3}\lrcorner {X_2}\lrcorner {X_1}\lrcorner \Omega \limits ^5$ conjugated to $V$ ($\Omega \limits ^5$ is a fixed local volume form on $M$), the loci of singular behavior of $V:{M_{\deg }}(V) = \{ p \in M|\omega (p) = 0\}$ and ${M_{{\mathrm {sing}}}}(V) = \{ p \in M|\omega \wedge {(d\omega )^2}(p) = 0\}$ are handled. The local classification of this pair of sets is carried out (outside a curve and a discrete set in ${M_{\deg }}(V)$) up to a smooth diffeomorphism. In the most complicated case, around points of a codimension 3 submanifold of $M$, ${M_{{\mathrm {sing}}}}(V)$ turns out to be diffeomorphic to the Cartesian product of ${\mathbb {R}^2}$ and the Whitney’s umbrella in ${\mathbb {R}^3}$.