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.
Categories: math.DS