Transversals to the flow induced by a differential equation on compact orientable $2$-dimensional manifolds

Abstract:

Every treatment of the theory of differential equations on a torus uses the fact that given a differential equation on a torus of class ${C^k}$, there is a non-null-homotopic closed Jordan curve $\Gamma$ of class ${C^k}$ which is transverse to the trajectories of the differential equation that pass through points of $\Gamma$. Such a curve necessarily cannot separate the torus. Here, we prove that given a differential equation on an n-fold torus ${T_n}$ of class ${C^k}$, possessing only “simple” singularities of negative index there is a non-null-homotopic closed Jordan curve $\Gamma$ of class ${C^k}$ which is a transversal. The nonseparating property, however, does not follow immediately. For the particular case ${T_2}$, we prove the existence of such a transversal that does not separate ${T_2}$.