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

Author:
Carl S. Hartzman

Journal:
Trans. Amer. Math. Soc. **167** (1972), 359-368

MSC:
Primary 34C40

MathSciNet review:
0294811

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 , there is a non-null-homotopic closed Jordan curve of class which is transverse to the trajectories of the differential equation that pass through points of . Such a curve necessarily cannot separate the torus. Here, we prove that given a differential equation on an *n*-fold torus of class , possessing only ``simple'' singularities of negative index there is a non-null-homotopic closed Jordan curve of class which is a transversal. The nonseparating property, however, does not follow immediately. For the particular case , we prove the existence of such a transversal that does not separate .

**[1]**Earl A. Coddington and Norman Levinson,*Theory of ordinary differential equations*, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. MR**0069338****[2]**A. Denjoy,*Sur les courbes définies par les équations différentielles à la surface du tore*, J. Math. Pures Appl. (9)**11**(1932), 333-375.**[3]**Philip Hartman,*Ordinary differential equations*, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR**0171038****[4]**John W. Milnor,*Topology from the differentiable viewpoint*, Based on notes by David W. Weaver, The University Press of Virginia, Charlottesville, Va., 1965. MR**0226651****[5]**V. V. Nemyckiĭ and V. V. Stepanov,*Kačestvennaya Teoriya Differencial′nyh Uravneniĭ*, OGIZ, Moscow-Leningrad,], 1947 (Russian). MR**0029483****[6]**Arthur J. Schwartz,*A generalization of a Poincaré-Bendixson theorem to closed two-dimensional manifolds*, Amer. J. Math. 85 (1963), 453-458; errata, ibid**85**(1963), 753. MR**0155061****[7]**J. J. Stoker,*Differentiable geometry*, Pure and Appl. Math., vol. 20, Interscience, New York, 1969. MR**39**#2072.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
34C40

Retrieve articles in all journals with MSC: 34C40

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1972-0294811-0

Keywords:
Differential equations,
manifolds,
transversal curves

Article copyright:
© Copyright 1972
American Mathematical Society