Abstract
The following theorem is proved for a closed manifold M with an oriented foliated structure of codimension 1 without limit cycles, supplemented by a foliation of one-dimensional normals: if every normal in M intersects every leaf, the same is true of the induced foliation on M (a universal covering of M).
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S. P. Novikov, “Topology of foliations,” Trudy Mosk. Matem. O-va,14, 248–278 (1965).
D. V. Anosov, “Geodesic flow on closed Riemann manifolds with negative curvature,” Trudy Matem. In-ta Acad. Nauk SSSR,90, 3–209 (1962).
A. Haefliger, “Variétés feuilletés,” Ann. Ec. Norm. Sup. Pisa (3),16, No. 4, 367–397 (1962).
R. Sacksteder, “Foliations and pseudogroups,” Amer. J. Math.,87, No. 1, 79–102 (1965).
A. J. Schwartz, “A generalization of a Poincaré-Bendixon theorem to closed two-dimensional manifolds,” Amer. J. Math.,85, No. 3, 453–458 (1963).
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Translated from Matematicheskie Zametki, Vol. 9, No. 2, pp. 181–191, February, 1971
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Brakhman, A.L. Foliation without limit cycles. Mathematical Notes of the Academy of Sciences of the USSR 9, 107–112 (1971). https://doi.org/10.1007/BF01316989
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DOI: https://doi.org/10.1007/BF01316989