Abstract
The survey is based on works on the theory of foliations reviewed in RZhMatematika during 1970–1979. The basic topics are the classification of foliations, characteristic classes, the qualitative theory of foliations (holonomy, growth of leaves, etc.), and special classes of foliations (compact foliations, Riemannian foliations, etc.).
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M. Cohen, “Approximation of foliations,” Can. Math. Bull.,14, No. 3, 311–314 (1971).
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M. Craioveanu, “Sur les sous-feuilletages d'une structure feuilletee,” C. R. Acad. Sci.,272, No. 11, A731-A733 (1971).
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M. Craioveanu, “Varietes banachiques feuilletees, II,” An. Univ. Timisoara, Ser. Sti. Mat.,13, No. 1, 11–12 (1975).
A. Davis and F. W. Wilson, “Vector fields tangent to foliations. I. Reeb foliations,” J. Diff. Eq.,11, No. 3, 491–498 (1972).
K. Decesaro and T. Nagano, “On compact foliations,” Proc. Symp. Pure Math., Stanford, Calif., 1973, Providence, Vol. 27, Part 1 (1975), pp. 277–281.
N. Desolneux-Moulis, “Sur certaines familles a un parametre de T2 × S2,” C. R. Acad. Sci.,AB287, No. 15, A1043-A1046 (1978).
P. R. Dippolito, “Codimension one foliations of closed manifolds,” Ann. Math.,107, No. 3, 403–453 (1978).
Tong van Duc, “Fibres vectoriels feuilletes,” Boll. Unione Mat. Ital.,A15, No. 1, 52–60 (1978).
Tong van Duc, “Fibres vectoriels feuilletes,” Kodai Math. J.,1, No. 2, 205–212 (1978).
T. Duchamp and M. Kalka, “Holomorphic foliations and the Kobayashi metric,” Proc. Am. Math. Soc.,67, No. 1, 117–122 (1977).
A. H. Durfee, “Foliations of odd-dimensional spheres,” Ann. Math.,96, No. 2, 407–411 (1972).
A. H. Durfee and H. B. Lawson, “Fibered knots and foliations of highly connected manifolds,” Invent. Math.,17, No. 3, 203–215 (1972).
R. Edwards, “A question concerning compact foliations,” Lect. Notes Math.,468, 2–4 (1975).
R. Edwards, K. Millett, and D. Sullivan, “Foliations with all leaves compact,” Topology,16, No. 1, 13–32 (1977).
D. B. A. Epstein, “The simplicity of certain groups of homeomorphisms,” Compos. Math.,22, No. 2, 165–173 (1970).
D. B. A. Epstein, “Periodic flows on three-dimensional manifolds,” Ann. Math.,95, No. 1, 66–82 (1972).
D. B. A. Epstein, “Foliations with all leaves compact,” Lect. Notes Math.,468, 1–2 (1975).
D. B. A. Epstein, “Foliations with all leaves compact,” Ann. Inst. Fourier,26, No. 1, 265–282 (1976).
D. B. A. Epstein, “A topology for the space of foliations,” Lect. Notes Math.,597, 132–150 (1977).
D. B. A. Epstein, K. Millet, and D. Tischler, “Leaves without holonomy,” J. London Math. Soc.,16, No. 3, 548–552 (1977).
D. B. A. Epstein and H. Rosenberg, “Stability of compact foliations,” Lect. Notes Math.,597, 151–160 (1977).
D. B. A. Epstein and E. Vogt, “A counterexample to the periodic orbit conjecture in codimension 3,” Ann. Math.,108, No. 3, 539–552 (1978).
E. Fedida, “Sur les feuilletages de Lie,” C. R. Acad. Sci.,272, No. 15, A999-A1001 (1971).
E. Fedida, “Structures differentiables sur le branchement simple et equations differentielles dans le plan,” C. R. Acad. Sci.,276, No. 26, A1657-A1659 (1973).
E. Fedida, “Sur l'existence des feuilletages de Lie,” C. R. Acad. Sci.,A278, No. 12, 835–837 (1974).
E. Fedida, “Sur la theorie des feuilletages associee au repere mobile: cas des feuilletages de Lie,” Lect. Notes Math.,652, 183–195 (1978).
J. M. Franks, “Two foliations in the plane,” Proc. Am. Math. Soc.,58, 262–264 (1976).
J. M. Franks, “Holonomy invariant cochains for foliations,” Proc. Am. Math. Soc.,62, No. 1, 16l-164 (1977).
M. Freeman, “Local complex foliation of real submanifolds,” Math. Ann.,209, No. 1, 1–30 (1974).
M. Freeman, “The Levi form and local complex foliations,” Proc. Am. Math. Soc.,57, No. 2, 369–370 (1976).
D. B. Fuchs, “Non-trivialite des classes caracteristiques de ε-structures. Applications aux classes caracteristiques des feuilletages,” C. R. Acad. Sci.,284, No. 17, A1017-A1019 (1977).
D. B. Fuchs, “Non-trivialite des classes characteristiques de ε-structures. Applications aux variations des classes caracteristiques de feuilletages,” C. R. Acad. Sci.,284, No. 18, A1105-A1107 (1977).
D. B. Fuchs, A. M. Gabrielov, and I. M. Gel'fand, “The Gauss-Bonnet theorem and the Atiyah-Patodi-Singer functionals for the characteristic classes of foliations,” Topology,15, No. 2, 165–188 (1976).
K. Fukui, “Codimension 1 foliations on simply connected 5-manifolds,” Proc. Jpn. Acad.,49, No. 6, 432–434 (1973).
K. Fukui, “An application of the Morse theory to foliated manifolds,” Nagoya Math. J.,54, 165–178 (1974).
K. Fukui, “On the foliated cobordisms of codimension-one foliated 3-manifolds,” Acta Hum. Sci. Univ., Sangio Kiotiensis Nat. Sci. Ser.,7, No. 4, 42–49 (1978).
P. M. D. Furness, “Affine foliations of codimension one,” Q. J. Math.,25, No. 98, 151–161 (1974).
P. M. D. Furness and E. Fedida, “Feuilletages transversalement affines de codimension 1,” C. R. Acad. Sci.,282, No. 16, A825-A827 (1976).
M. Garancon, “Homotopie et holonomie de certains feuilletages de codimension 1,” Ann. Inst. Fourier,22, No. 2, 61–71 (1972).
M. Garancon, “Feuilletages transversalement analytiques de codimension 1 admettant une transversale fermee qui coupe toutes les feuilles,” Ann. Inst. Fourier,22, No. 4, 271–287 (1972).
R. B. Gardner, “Differential geometry and foliations: the Godbillon-Vey invariant and the Bott-Pasternack vanishing theorems,” Lect. Notes Math.,652, 75–94 (1978).
D. Gauld, “Foliations on topological manifolds,” Math. Chron.,2, Nos. 1–2, 29–41 (1972).
C. Godbillon, “Feuilletages ayant la propriete du prolongement des homotopies,” These Doct. Sci. Math., Fac. Sci. Univ. Strasbourg (1967).
C. Godbillon, “Problemes d'existence et d'homotopie dans les feuilletages,” Lect. Notes Math.,244, 167–181 (1971).
C. Godbillon, “Fibres en droites et feuilletages du plan,” Enseign. Math.,18, Nos. 3–4, 213–224 (1972-973).
C. Godbillon, “Cohomologies d'algebres de Lie de champs de vecteurs formels,” Lect. Notes Math.,383, 69–87 (1974).
C. Godbillon, “Feuilletages de Lie,” Lect. Notes Math.,392, 10–13 (1974).
C. Godbillon, “Invariants of foliations,” Global Anal. Appl. Lect. Int. Semin. Course Trieste, Vienna, Vol. 2 (1974), pp. 215–219.
C. Godbillon and J. Vey, “Un invariant des feuilletages de codimension 1,” C. R. Acad. Sci.,273, No. 2, A92-A95 (1971).
B. F. Golbus, “On the singularities of foliations and of vector bundle maps,” Bol. Soc, Brasil. Mat.,7, No. 1, 11–35 (1976).
B. F. Golbus, “On extending local foliations,” Q. J. Math.,28, No. 110, 163–176 (1977).
R. Goldman, “The holonomy ring on the leaves of foliated manifolds,” J. Diff. Geom.,11, No. 9, 411–449 (1976).
S. Goodman, “Closed leaves in foliated 3-manifolds,” Proc. Nat. Acad. Sci. USA,71, No. 11, 4414–4415 (1974).
S. Goodman, “Closed leaves in foliations of codimension one,” Comment, Math. Helv.,50, No. 3, 383–388 (1975).
S. Goodman, “On the structure of foliated 3-manifolds separated by a compact leaf,” Invent. Math.,39, No. 3, 213–221 (1977).
S. Guelorget, “Algebre de Weil du groupe lineare. Application aux classes caracteristiques d'un feuilletage,” Lect. Notes Math.,484, 179–191 (1975).
S. Guelorget and G. Joubert, “Algebre de Weil et classes caracteristiques d'un feuilletage,” C. R. Acad. Sci.,277, No. 1, A11-A14 (1973).
A. Haefliger, “Sur les feuilletages analytiques,” C. R. Acad. Sci.,242, No. 25, 2908–2910 (1956).
A. Haefliger, “Feuilletages sur les varietes ouvertes,” Topology,9, No. 2, 183–194 (1970).
A. Haefliger, “Homotopy and integrability,” Lect. Notes Math.,197, 133–163 (1971).
A. Haefliger, “Teorema de Bott sobre una obstruccion topologica a la integrabilidad complota,” Rev. Mat. Hisp.-Amer.,32, Nos. 1–2, 21–32 (1972).
A. Haefliger, “Sur les classes caracteristiques des feuilletages,” Lect. Notes Math.,317, 239–260 (1973).
A. Haefliger, “Cohomology of Lie algebras and foliations,” Lect. Notes Math.,652, 1–12 (1978).
G. Hector, “Sur le type des feuilletages transyerses de R3,” C. R. Acad. Sci.,273, No. 18, A810-A813 (1971).
G. Hector, “Ouverts incompressibles et theoreme de Denjoy-Poincare pour les feuilletages,” C. R. Acad. Sci.,274. No. 2, A159-A162 (1972).
G. Hector, “Ouverts incompressibles et structure des feuilletages de codimension 1,” C. R. Acad. Sci.,274, A741-A744 (1972).
G. Hector, “Sur les feuilletages presque sans holonomie,” C. R. Acad. Sci.,274, No. 24, A1703-A1706 (1972).
G. Hector, “Actions de groupes de diffeomorphismes de {0,1},n” Lect. Notes Math.,392, 14–22 (1974).
G. Hector, “Quelques exemples de feuilletages especes rares,” Ann. Inst. Fourier,26, No. 1, 239–264 (1976).
G. Hector, “Feuilletages en cylindres,” Lect. Notes Math.,597, 252–270 (1977).
G. Hector, “Leaves whose growth is neither exponential nor polynomial,” Topology,16, No. 4, 451–459 (1977).
G. Hector, “Croissance des feuilletages presque sans holonomie,” Lect. Notes Math.,652, 141–182 (1978).
J. L. Heitsch, “Deformations of secondary characteristic classes,” Topology,12, No. 4, 381–388 (1973).
J. L. Heitsch, “A eohomology for foliated manifolds,” Bull. Am. Math. Soc.,79, No. 6, 1283–1285 (1973).
J. L. Heitsch, “A eohomology for foliated manifolds,” Comment. Math. Helv.,50, No. 2, 197–218 (1975).
J. L. Heitsch, “A remark on the residue theorem of Bott,” Indiana Univ. Math. J.,25, No. 12, 1139–1147 (1976).
J. L. Heitsch, “Residues and characteristic classes of foliations,” Bull. Am. Math. Soc.,83, No. 3, 397–399 (1977).
J. L. Heitsch, “Independent variations of secondary classes,” Ann. Math.,108, No. 3, 421–460 (1978).
J. L. Heitsch, “Derivatives of secondary characteristic classes,” J. Diff. Geom.,13, 311–339 (1978).
M. R. Hermann, “The Godbillon-Vey invariant of foliations by planes of T3,” Lect. Notes Math.,597, 294–307 (1977).
M. Hirsch, “Foliated bundles, flat manifolds and invariant measures,” Lect. Notes Math.,468, 8–9 (1975).
M. Hirsch, “A stable analytic foliation with only exceptional minimal sets,” Lect. Notes Math.,468, 9–10 (1975).
M. Hirsch and W. P. Thurston, “Foliated bundles, invariant measures and flat manifolds,” Ann. Math.,101, No. 3, 369–390 (1975).
H. Holmann, “Holomorphe Blätterungen komplexen Räume,” Comment. Math. Helv.,47, No. 2, 185–204 (1972).
H. Holmann, “On the stability of holomorphic foliations with all leaves compact,” Lect. Notes Math.,683, 217–248 (1978).
G. Ikegami, “Existence of regular coverings associated with leaves of codimension one foliations,” Nagoya Math. J.,67, 15–34 (1977).
Hideki Imanishi, “Sur l'existence des feuilletages S1-invariants,” J. Math. Kyoto Univ.,12, No. 2, 297–305 (1972).
T. Inaba, “On stability of proper leaves of codimension one foliations,” J. Math. Soc. Jpn.,29, No. 4, 771–778 (1977).
S. Jekel, “On two theorems of A. Haefliger concerning foliations,” Topology,15, No. 3, 267–271 (1976).
S. Jekel, “Simplicial K(G, 1)'s,” Manuscr. Math.,21, No. 2, 189–203 (1977).
S. Jekel, “A note on the perfection of the fundamental group of the classifying space for codimension one real analytic foliations,” Bol. Soc. Mat. Max.,22, No. 2, 58–59 (1977).
D. L. Johnson and L. B. Witt, “Totally geodesic foliations on 3-manifolds,” Proc. Am. Math. Soc.,76, No. 2, 355–357 (1979).
J. P. Jouanolou, “Feuilles compactes des feuilletages algebriques,” Math. Ann.,241, No. 1, 69–72 (1979).
G. Joubert, “Estructuras foliadas,” Rev. Colomb. Mat.,2, No. 4, 105–116 (1968).
G. Joubert and R. Moussu, “El teorema de Novikov,” Rev. Colomb. Mat.,2, No. 4, 117–123 (1968).
G. Joubert and R. Moussu, “Feuilletage sans holonomie d'une variete fermee,” C. R. Acad. Sci.,270, No. 8, A507-A509 (1970).
G. Joubert, R. Moussu, and D. Tischler, “Sur les classes caracteristiques des feuilletages produits,” C. R. Acad. Sci.,275, No. 3, A171-A174 (1972).
F. W. Kamber and Ph. Tondeur, “Cohomologie des algebres de Weil relatives tronquees,” C. R. Acad. Sci.,276, No. 6, A459-A462 (1973).
F. W. Kamber and Ph. Tondeur, “Algebres de Weil semi-simpliciales,” C. R. Acad. Sci.,276, No. 18, A1177-A1179 (1973).
F. W. Kamber and Ph. Tondeur, “Homomorphisme caracteristique d'un fibre principal feuillete,” C. R. Acad. Sci.,276, No. 21, A1407-A1410 (1973).
F. W. Kamber and Ph. Tondeur, “Classes caracteristiques derivees d'un fibre principal feuillete,” C. R. Acad. Sci.,276, No. 22, A1449-A1452 (1973).
F. W. Kamber and Ph. Tondeur, “Characteristic invariants of foliated bundles,” Manuscr. Math.,11, No. 1, 51–89 (1974).
F. W. Kamber and Ph. Tondeur, “Classes caracteristiques generalisees des fibres feuilletes localement homogenes,” C. R. Acad. Sci.,A279, No. 25, 847–850 (1974).
F. W. Kamber and Ph. Tondeur, “Quelques classes caracteristiques generalisees non triviales de fibres feuilletes,” C. R. Acad. Sci.,A279, No. 26, 921–924 (1974).
F. W. Kamber and Ph. Tondeur, “Semisimplicial Weil algebras and characteristic classes for foliated bundles in Cech cohomology,” Proc. Symp. Pure Math., Stanford, Calif., 1973, Providence, Vol. 27, Part 1 (1975), pp. 283–294.
F. W. Kamber and Ph. Tondeur, “Foliated bundles and characteristic classes,” Lect. Notes Math.,493, 207 (1975).
F. W. Kamber and Ph. Tondeur, “Classes caracteristiques et suite spectrales d'Eilenberg-Moore,” C. R. Acad. Sci.,283, No. 12, A883-A886 (1976).
F. W. Kamber and Ph. Tondeur, “G-foliations and their characteristic classes,” Bull. Am. Math. Soc.,84, No. 6, 1086–1124 (1978).
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B. Kaup, “Ein geometrisches Endlichkeitskriterium für Untergruppen von Aut(C, 0) und holomorphe 1-codimensionale Blätterungen,” Comment. Math. Helv.,53, No. 2, 295–299 (1978).
U. Koschorke, “Singularities and bordisms of q-plane fields and of foliations,” Bull. Am. Math. Soc.,80, No. 4, 760–765 (1974).
U. Koschorke, “Line fields transversal to foliations,” Proc. Symp. Pure Math., Stanford, Calif., 1973, Providence, Vol. 27, Part 1 (1975), pp. 295–301.
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C. Lamoureux, “Feuilletages de codimension 1. Holonomie et homotopie,” C. R. Acad. Sci.,270, No. 26, A1718-A1721 (1970).
C. Lamoureux, “Une condition pour qu'une feuille soit propre et ait une enveloppe composee de feuilles fermees,” C. R. Acad. Sci.,274, No. 1, A31-A34 (1972).
G. Lamoureux, “Feuilles fermees et captage; applications,” C. R. Acad. Sci.,277, No. 13, A579-A581 (1973).
C. Lamoureux, “Feuilles exceptionelles, feuilles denses, homologie et captage,” C. R. Acad. Sci.,277, No. 21, A1041-A1043 (1973).
C. Lamoureux, “Sur quelques phenomenes de captage,” Ann. Inst. Fourier,23, No. 4, 229–243 (1973–1974).
C. Lamoureux, “The structure of foliations without holonomy on non-compact manifolds with fondamental group Z,” Topology,13, No. 3, 219–224 (1974).
C. Lamoureux, “Quelques proprietes globales des feuilletages par des feuilles simplement connexes non-necessairement bornees,” C. R. Acad. Sci.,A279, No. 23, 813–816 (1974).
C. Lamoureux, “Groupes d'homologie et d'homologie d'ordre superieur des varietes compactes ou non compactes feuilletees en codimension 1,” C. R. Acad. Sci.,280, No. 7, A4H-A414 (1975).
C. Lamoureux, “Non-bounded leaves in codimension one foliations,” Lect. Notes Math.,484, 257–272 (1975).
C. Lamoureux, “Feuilles non captees et feuilles denses,” Ann. Inst. Fourier,25, No. 2, 285–293 (1975).
C. Lamoureux, “Feuilletages des varietes compactes et non compactes,” Ann. Inst. Fourier,26, No. 2, 221–271 (1976).
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R. Langevin and H. Rosenberg, “On stability of compact leaves and fibrations,” Topology,16, No. 1, 107–111 (1977).
R. Langevin and H. Rosenberg, “Integrable perturbations of fibrations and a theorem of Scifert,” Lect. Notes Math.,652, 122–127 (1978).
F. Laudenbach and R. Roussarie, “Un exemple de feuilletage sur S2,” Topology,9, No. 1, 63–70 (1970).
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H. B. Lawson, “Codimension-one foliations of spheres,” Ann. Math.,94, No. 3, 494–503 (1971).
H. B. Lawson, “Foliations,” Bull. Am. Math. Soc.,80, No. 3, 369–418 (1974).
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C. Lazarov and J. S. Pasternack, “Residues and characteristic classes for Riemannian foliations,” J. Diff. Georn.,11, No. 4, 599–612 (1976).
C. Lazarov and H. Shulman, “Obstructions to foliation preserving Lie group actions,” Topology,18, No. 3, 255–256 (1979).
D. Lehmann, “I-homotopie dans les espaces de connexions et classes exotiques de Chern-Simons,” C. R. Acad. Sci.,275, No. 18, A835-A838 (1972).
D. Lehmann, “Classes caracteristiques exotiques et J-connesite des espases de connexions,” Ann. Inst. Fourier,24, No. 3, 267–300 (1974).
D. Lehmann, “Sur l'approximation de certains feuilletages nilpotents par des fibrations,” C. R. Acad. Sci.,AB286, No. 5, A251-A254 (1978).
D. Lehmann, “Sur la generalization d'un theoreme de Tischler a certains feuilletages nilpotents,” Proc. K. Ned. Akad. Wet., Ser. A,82, No. 2, 177–189 (1979).
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C.-C. Liang, “Multifouations on open manifolds,” Math. Ann.,221, No. 2, 143–146 (1976).
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L. Lininger, “Codimension 1 foliations on manifolds with even index,” Lect. Notes Math.,438, 336–338 (1975).
J. Martinet, “Classes caracteristiques des systemes de Pfaff,” Lect. Notes Math.,392, 30–36 (1974).
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J. N. Mather, “On Haefliger's classifying space. I,” Bull. Am. Math. Soc.,77, No. 6, 1111–1115 (1971).
J. N. Mather, “The vanishing of the homology of certain groups of homeomorphisms,” Topology,10, No. 4, 297–298 (1971).
J. N. Mather, “Integrability in codimension 1,” Comment. Math. Helv.,48, No. 2, 195–233 (1973).
J. N. Mather, “Simplicity of certain groups of diffeomorphisms,” Bull. Am. Math. Soc.,80, No. 2, 271–273 (1974).
J. N. Mather, “Commutators of diffeomorphisms,” Comment. Math. Helv.,49, No. 4, 512–528 (1974).
J. N. Mather, “Commutators of diffeomorphisms. II,” Comment. Math. Helv.,50, No. 1, 33–40 (1975).
J. N. Mather, “Foliations and homology of groups of diffeomorphisms,” Proc. Int. Congr. Math., Vancouver, 1974, Vol. 2, S. 1 (1975), pp. 35–37.
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A. Medina, “Quelques remarques sur les feuilletages affines,” C. R. Acad. Sci.,282, No. 19, A1159-A1162 (1976).
A. Medina, “Quelques remarques sur les feuilletages affines,” Cah. Topol. Geom. Different., Ch. Ehresmann,17, No. 1, 59–68 (1976).
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K. Millett, “Compact foliations,” Lect. Notes Math.,484, 277–287 (1975).
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T. Mizutani, “Foliated cobordisms of S3 and examples of foliated 4-manifolds,” Topology,13, No. 4, 353–362 (1974).
T. Mizutani, “Foliations and foliated cobordisms of spheres in codimension one,” J. Math. Soc. Jpn.,27, No. 2, 264–280 (1975).
T. Mizutani and T. Tsuboi, “Foliations without holonomy and foliated bundles,” Sci. Rep. Saitama Univ., Ser. A,9, No. 2, 45–55 (1979).
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P. Molino, “Classe d'Atiyah d'un feuilletage et connexions transverses projectables,” C. R. Acad. Sci.,272, No. 12, A779-A781 (1971).
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P. Molino, “La classe d'Atiyah d'un feuilletage comme cocycle de deformation infinitesimale,” C. R. Acad. Sci.,A278, No. 10, 719–721 (1974).
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P. Molino, “Sur la geometrie transverse des feuilletages,” Ann. Inst. Fourier,25, No. 2, 279–284 (1975).
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J. F. Plante, “Foliations with measure preserving holonomy,” Ann. Math.,102, No. 2, 327–361 (1975).
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O. H. Rasmussen, “Reeb foliations,” Prepr. Ser. Mat. Inst. Aarhus Univ., No. 58, 8 (1973).
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P. A. Schweitzer, “Some problems in foliation theory and related areas,” Lect. Notes Math.,652, 240–252 (1978).
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Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 18, pp. 151–213, 1981.
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Fuks, D.B. Foliations. J Math Sci 18, 255–291 (1982). https://doi.org/10.1007/BF01255616
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DOI: https://doi.org/10.1007/BF01255616