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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

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


Author: Carl S. Hartzman
Journal: Trans. Amer. Math. Soc. 167 (1972), 359-368
MSC: Primary 34C40
MathSciNet review: 0294811
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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}$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0294811-0
PII: S 0002-9947(1972)0294811-0
Keywords: Differential equations, manifolds, transversal curves
Article copyright: © Copyright 1972 American Mathematical Society