Characteristic foliations of spheres embedded in the standard overtwisted structure \((\mathbb{R}^3,\zeta_1)\)
Let \((M,\xi)\) be a 3-contact manifold, and \(S\hookrightarrow (M,\xi)\) a closed, oriented \(C^\infty\) surface embedded in \(M\). Then there is an oriented 1-foliation \({\mathcal F}= S_\xi\) induced by \(\xi\) on \(S\), which is called the characteristic foliation. The \((M,\xi)\) is called overtwisted if there exists a 2-disc \(D\) embedded in \(M\) such that \(\partial D\) is tangent to the distribution non-integrable \(\xi\). Let \((\mathbb{R}^3, \zeta_1)\) be the standard overtwisted contact structure given by: \(\cos\rho dz-\rho \sin\rho d \varphi=0\), where \(\rho,\varphi,z\) are cylindric coordinates. The aim of this paper is to show that any Morse-Smale foliation on the sphere with null Euler class, is up to isotopy, the characteristic foliation of a sphere of \((\mathbb{R}^3,\zeta_1)\). The fundamental result is the following Theorem. Every Morse-Smale 1-foliation on the sphere \(S^2\) with null Euler class, is the characteristic foliation of a sphere embedded in \((\mathbb{R}^3,\zeta_1)\). The method is constructive.
scientific article; zbMATH DE number 1383254