Principal cycles of one dimensional foliations associated to a plane field in $\mathbb{E}^3$

Abstract

In this work it will be analyzed η-principal cycles (compact leaves) of one dimensional singular foliations associated to a plane field ∆ η defined by a unit and normal vector field η in E 3 . The leaves are orthogonal to the orbits of η and are the integral curves corresponding to directions of extreme normal curvature of the plane field ∆ η . It is shown that, generically, given a η-principal cycle it can be make hyperbolic (the derivative of the first return of the Poincaré map has all eigenvalues disjoint from the unit circle) by a small deformation of the vector field η. Also is shown that for a dense set of unit vector fields, with the weak C r -topology of Whitney, the η-principal cycles are hyperbolic.