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 $Δ_η$ defined by a unit and normal vector field $η$ in $ \mathbb 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 $Δ_η$. % 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.