Transversals to the flow induced by a differential equation on compact orientable $2$-dimensional manifolds
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- by Carl S. Hartzman PDF
- Trans. Amer. Math. Soc. 167 (1972), 359-368 Request permission
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 ${T_2}$.References
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Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 167 (1972), 359-368
- MSC: Primary 34C40
- DOI: https://doi.org/10.1090/S0002-9947-1972-0294811-0
- MathSciNet review: 0294811