Abstract
The holonomy of a leaf is a mapping between the transversals to the leaf. If this mapping is the identity then the leaf is said to be “without” holonomy. When all leaves of a foliation satisfy such a property then F is called a foliation without holonomy. Closed 1—forms define a class of orientable foliations without holonomy. Holomorphic quadratic differentials define a class of non orientable foliations without holonomy. The converse statements are true as well.
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Bibliographic Notes
Aranson, S. H. and Zhuzhoma, E. V., 1994 About structure of quasiminimal sets of foliations on surfaces, Mat. Sbornik 185, no.8, 31–62, Translation: Mat. Sbornik.
Cherry, T., 1938 Analytic quasi-periodic curves of discontinuous type on torus, Proc. London Math. Soc. 44, 175–215.
Denjoy, A., 1932 Sur les courbes définies par les équations différentielles à la surface du tore, J. Math. Pures et Appl. (ser 9) 11, 333–375.
Fathi, A., Laudenbach, F., and Poénaru, Z., 1979 Travaux de Thurston sur les surfaces, Séminaire Orsay, France, Astérisque 66–67.
Gardiner, C. J., 1985 The structure of flows exhibiting nontrivial recurrence on two-dimensional manifolds, J. Differential Equations 57, 138–158.
Guttiérrez, C., 1986 Smoothing continuous flows on 2-manifolds and recurrences, Ergod. Th. and Dyn. Syst. 6, 17–44.
Levitt, G., 1982 Pantalons et feuilletages des surfaces, Topology 21, 9–33.
Levitt, G., 1982 Feuilletages des surfaces, Ann. Inst. Fourier 32, 179–217.
Levitt, G., 1983 Flots topologiquement transitifs sur les surfaces compactes sans bord: contrexemple à une conjecture de Katok, Ergod. Th. and Dynam Sys. 3, 241–249.
Levitt, G., 1983 Feuilletages des surfaces, Thèse, Paris.
Levitt, G., 1987 La décomposition dynamique et la différentiabilité des feuilletages des surfaces, Ann. Inst. Fourier 37, 85–116.
Maier, A. G., 1939 De trajectoires sur les surfaces orientées, C.R. Acad. Sci. URSS 24, 673–675.
Maier, A. G., 1943 About trajectories on orientable surfaces, Mat. Sbornik 12, no. 1, 71–84.
Markley, N., 1969 The Poincaré-Bendixson theorem for the Klein bottle, Trans. Amer. Math. Soc. 135, 159–165.
Neumann, D., 1978 Smoothing continuous flows on 2-manifolds, J. Differential Equations 28, 327–344.
Novikov, S. P., 1965 Topology of foliations, Trudy Mosk. Mat. Obshch. 14, 248–278.
Plykin, R. V., 1974 Sourses and sinks of A-diffeomorphisms on surfaces, Mat. Sbornik 94, 243–264, Translation: Mat. Sbornik.
Rosenberg, H., 1983 Labyrinths in the disc and surfaces, Ann. of Math. 117, 1–33.
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© 2001 Springer-Verlag Berlin Heidelberg
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Nikolaev, I. (2001). Foliations Without Holonomy. In: Foliations on Surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol 41. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04524-4_4
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DOI: https://doi.org/10.1007/978-3-662-04524-4_4
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