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 Poincaré 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.

References in corpus (1)

• The curvature of contact structure on 3-manifolds