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

Abstract

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