On pairs of regular foliations in $${\mathbb{R}^3}$$ and singularities of map-germs

Abstract

We study germs of pairs of codimension one regular foliations in R 3 . We show that the discriminant of the pair determines the topological type of the pair. We also consider various classifications of the singularities of the discriminant.

PROJECTIONS OF HYPERSURFACES IN ℝ4 WITH BOUNDARY TO PLANES

“…Theorem 1.1 models codimension less than or equal to four simple singularities of the discriminant map-germ up to subgroup B. The B-class of a discriminant map-germ determines if the associated pair of foliations is topologically equivalent to one of the discrete topological models given in [10]. Indeed, suppose Table 1.…”

“…In local coordinates, the discriminant is given by the fibre of a map-germ F : ‫ޒ(‬ 3 , 0) → ‫ޒ(‬ 2 , 0), which we call the discriminant map-germ. In [10] the authors show that the discriminant D(ω, η) determines the local topological type of the pair (ω, η) and obtain a complete list of discrete topological models (Theorem 4.1, p. 108). Theorem 1.1 models codimension less than or equal to four simple singularities of the discriminant map-germ up to subgroup B.…”

A Note on Pairs of Regular Foliations on the Solid Torus

“…This is generically a germ of a space curve. In [4] the authors show that the discriminant D(ω, η) determines the local topological type of the pair (ω, η) and obtain a complete list of discrete topological models. Because the discriminant plays a key role in Received: January 20, 2014 c 2014 Academic Publications the topological classification, in [4] and [5] the authors analyze its singularities considering the discriminant, in local coordinates, given by the fibre of a mapgerm F :…”

“…Also if the discriminant, in local coordinates, is given by the fibre of a mapgerm F : (R 3 , 0) → (R 2 , 0) and F has a finitely K * -determined singularity, then the discriminant is transverse, away from the origin, to the foliation given by discs D θ , θ ∈ S 1 . See [4] for various classifications of the singularities of the discriminant.…”

“…The motivation for this work came from [4], which is a generalization of [7] for pairs of germs or regular foliations in the plane. Pairs of foliations appear in, and have applications to, control theory, partial differential equations and differential geometry (see for example [3]).…”