Skip to main content
SpringerLink
Log in

Foliations with Leaf Structures

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

REFERENCES

  1. R. Abraham and J. E. Marsden, Foundations of Mechanics, Reading, Mass. (1978).

  2. B. N. Apanasov, Geometry of Discrete Groups and Manifolds [in Russian], Nauka, Moscow (1980).

    Google Scholar 

  3. R. Barre, "Theorie des Q-varietes differentielles et analytiques," Ann. Inst. Fourier, Grenoble, 23, No. 3, 227-312 (1973).

    Google Scholar 

  4. I. V. Bel'ko, "The Atiyah-Molino class of fibered Lie algebroids," Dokl. Akad. Nauk Belarusi, 37, No. 5, 16-18 (1993).

    Google Scholar 

  5. R. A. Blumenthal and J. J. Hebda, "Ehresmann connections for foliations," Indiana Math. J., 33, No. 4, 597-612 (1984).

    Google Scholar 

  6. R. A. Blumenthal and J. J. Hebda, "Complementary distributions which preserve the leaf geometry and applications to totally geodesic foliations," Quart. J. Math., 35, No. 4, 597-612 (1984).

    Google Scholar 

  7. R. A. Blumenthal and J. J. Hebda, "An analogue of the holonomy bundle for a foliated manifold," Tôhoku Math. J., 40, No. 2, 189-197 (1988).

    Google Scholar 

  8. R. Bott and L. W. Tu, Differential Forms in Algebraic Topology [Russian translation], Nauka, Moscow (1989).

    Google Scholar 

  9. F. Brickell and R. S. Clark, "Integrable almost tangent structures," J. Differ. Geom., 9, No. 4, 557-563 (1974).

    Google Scholar 

  10. G. Chachelet and H. Rosenberg "Manifolds which admit Rn-actions," IHES Publ. Math., No. 43 (1973).

  11. L. Cordero and R. Wołak, "Examples of foliations with foliated geometrical structures," Pacif. J. Math., 142, No. 2, 265-276 (1990).

    Google Scholar 

  12. M. Crampin and G. Thompson, "Affine bundles and integrable almost tangent structures," Math. Proc. Cambridge Philos. Soc., 98 (1985).

  13. P. Dazord and G. Hector, "Integration symplectique des variétés de Poisson totalement asphériques," In: Symplectic Geometry, Groupoids, and Integrable Systems. Sém. Sud Rhod. à Berkeley, 1989, Springer (1991), pp. 38-72.

  14. D. Fried, W. Goldman, and M. Hirsch, "Affine manifolds with nilpotent holonomy," Comment. Math. Helv., 56, Fasc. 4, 487-523 (1981).

    Google Scholar 

  15. D. B. Fuks, "Foliations," In: Progress in Science and Technology, Series on Algebra, Topology, and Geometry [in Russian], Vol. 18, All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1981), pp. 151-212.

    Google Scholar 

  16. P. M. D. Furness, "Affine foliations of codimension one," Quart. J. Math., 25, No. 98 (1974).

  17. P. M. D. Furness and E. Fédida, "Transversely affine foliations," Glasgow Math. J., 17, Part 2, 106-112 (1976).

    Google Scholar 

  18. T. I. Gaisin, "The Spencer complex for manifolds over algebras," In: Tr. Geom. Sem., No. 23, Kazan (1997), pp. 33-41.

    Google Scholar 

  19. C. Godbillon, "Holonomie transversale," C. R. Acad. Sci., Ser. A, B, 264, No. 24, 1050-1052.

  20. W. Goldman and M. Hirsch, "Polynomial forms on affine manifolds," Pacific J. Math., 101, No. 1 (1982).

  21. W. Goldman, "On polynomial cohomology of affine manifolds," Invent. Math., 65, 453-457 (1982).

    Google Scholar 

  22. A. Haefliger, "Homotopy and integrability," In: Lect. Notes Math., Vol. 197 (1971), pp. 133-163.

    Google Scholar 

  23. G. Hector and U. Hirsch, Introduction to the Geometry of Foliations, Part A, B, Vieweg, Braunschweig (1981, 1983).

    Google Scholar 

  24. T. Inaba, The tangentially a.ne structure of Lagrangian foliations and the tangentially projective structure of Legendrian foliations, Preprint.

  25. T. Inaba and K. Masuda, "Tangentially affine foliations and leafwise affine functions on the torus," Kodai Math. J., 16, No. 1, 32-43 (1993).

    Google Scholar 

  26. T. Inaba and S. Matsumoto, "Some qualitative aspects of transversely projective foliations," Proc. Jpn. Acad., Ser. A, 65, No. 4, 116-118 (1989).

    Google Scholar 

  27. F. Kamber and Ph. Tondeur, "Characteristic classes of foliated bundles," In: Lect. Notes Math., Vol. 494 (1975).

  28. A. E. Kasimi-Alaoui, "Sur la cohomologie feuilletée," Compos. Math., 49, 195-205 (1983).

    Google Scholar 

  29. A. E. Kasimi-Alaoui and A. Tihami, "Cohomologie bigradu´ee de certains feuilletages," Bull. Soc. Math. Belg., Ser. B, 38, 144-156 (1986).

    Google Scholar 

  30. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry [Russian translation], Vols. 1, 2, Nauka, Moscow (1979, 1981).

    Google Scholar 

  31. K. Kodaira, Complex Manifolds and Deformations of Complex Structures, Springer (1986).

  32. K. Kodaira and D. C. Spencer, "Multifoliate structures," Ann. Math., 74, No. 1, 52-100 (1961).

    Google Scholar 

  33. I. Kolář, P. W. Michor, and J. Slovák, Natural Operations in Differential Geometry, Springer-Verlag (1993).

  34. B. Lawson, The Quantative Theory of Foliations, CBMS, No. 27, AMS, Providence (1977).

    Google Scholar 

  35. M. V. Losik, "On characteristic classes of structures on a manifold," Funkts. Anal. Prilozh., 21, No. 3, 38-52 (1987).

    Google Scholar 

  36. M. V. Losik, "On a certain generalization of a manifold and its characteristic classes," Funkts. Anal. Prilozh., 24, No. 1, 29-37 (1990).

    Google Scholar 

  37. M. A. Malakhal'tsev, "(X,G)-Foliations," Izv. Vuzov, Mat., No. 7, 55-65 (1996).

    Google Scholar 

  38. M. A. Malakhal'tsev, "The Lie derivative and cohomology of G-structures," Lobachevskii J. Math., 3, 197-200 (1999).

    Google Scholar 

  39. P. Molino, Riemannian Foliations, Birkhäuser (1988).

  40. P. Molino, "Sur la géométrie transverse des feuilletages," Ann. Inst. Fourier, 25, No. 2, 279-284 (1975).

    Google Scholar 

  41. P. Molino, "Théorie des G-structure: le problème d'equivalence," In: Lect. Notes Math., Vol. 588, Springer (1977).

  42. S. P. Novikov, "Topology of foliations," Tr. Mosk. Mat. Obshch., 14, 248-278 (1965).

    Google Scholar 

  43. J. F. Plante, "Locally free affine group actions," Trans. Amer. Math. Soc., 259, No. 2, 449-456 (1980).

    Google Scholar 

  44. J. Pommaret, Systems of Partial Differential Equations and Lie Pseudo-Groups [Russian translation], Mir, Moscow (1983).

    Google Scholar 

  45. M. N. Sabitova, "Polynomial manifolds," Tensor, 62, No. 3 (2000).

  46. E. Salem, "Riemannian foliations and pseudo-groups of isometries," In: P. Molino, Riemannian Foliations, Appendix D (1988).

  47. K. S. Sarkaria, "A finiteness theorem for foliated manifolds," J.Math. Soc. Jpn., 30, No. 4, 687-696 (1978).

    Google Scholar 

  48. B. N. Shapukov, "Connections on differentiable bundles," In: Progress in Science and Technology, Series on Problems in Geometry [in Russian], Vol. 15, All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1983), pp. 61-93.

    Google Scholar 

  49. A. P. Shirokov, "Structures on differentiable manifolds," In: Progress in Science and Technology, Series on Algebra, Topology, and Geometry [in Russian], Vol. 11, All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1974), pp. 153-207.

    Google Scholar 

  50. V. V. Shurygin, "Jet bundles as manifolds over algebras," In: Progress in Science and Technology, Series on Problems in Geometry [in Russian], Vol. 19, All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1987), pp. 3-22.

    Google Scholar 

  51. V. V. Shurygin, "Manifolds over algebras and their applications in geometry of jet bundles," Usp. Mat. Nauk, 48, No. 2, 75-106 (1993).

    Google Scholar 

  52. V. V. Shurygin, "Ehresmann connection for the canonical bundle over a local algebra," Mat. Zametki, 59, No. 2, 303-310 (1996).

    Google Scholar 

  53. V. V. Shurygin, "Manifolds on algebras that are equivalent to a jet bundle," Izv. Vuzov, Mat., No. 10, 68-79 (1992).

    Google Scholar 

  54. V. V. Shurygin, "On cohomologies of manifolds over local algebras," Izv. Vuzov, Mat., No. 9, 71-85 (1996).

    Google Scholar 

  55. E. Spanier, Algebraic Topology [Russian translation], Mir, Moscow (1971).

    Google Scholar 

  56. I. Tamura Topology of Foliations [Russian translation], Mir, Moscow (1979).

    Google Scholar 

  57. G. Thompson, "Integrable almost cotangent structures and Legendrian bundles," Math. Proc. Cambridge Philos. Soc., 101, Part 1, 61-78 (1987).

    Google Scholar 

  58. W. P. Thurston, The Geometry and Topology of 3-Manifolds, Mimeographed Lecture Notes. Princeton Univ. (1978/79).

  59. D. Tischler, "Locally free actions of Rn-1 on M n without compact orbits," Topology, 13, No. 3, 215-217 (1974).

    Google Scholar 

  60. V. V. Trofimov and A. T. Fomenko, Algebra and Geometry of Integrable Hamiltonian Differential Equations [in Russian], Faktorial, Moscow (1995).

    Google Scholar 

  61. I. Vaisman, "d f-Cohomologies of Lagrangian foliations," Monat. Math., 106, 221-244 (1988).

    Google Scholar 

  62. I. Vaisman, "Basics of Lagrangian foliations," Publ. Mat. Barc., 33, No. 3, 559-575 (1989).

    Google Scholar 

  63. I. Vaisman, Cohomology and Differential Forms, Marcel Dekker, New York (1973).

    Google Scholar 

  64. V. V. Vishnevskii, A. P. Shirokov, and V. V. Shurygin, Spaces over Algebras [in Russian], Kazan University, Kazan (1985).

    Google Scholar 

  65. H. E. Winkelnkemper, "The graph of the foliation," Ann. Glob. Anal. Geom., 1, No. 3, 51-75 (1983).

    Google Scholar 

  66. R. Wołak, Geometric structures on foliated manifolds, Preprint (1989).

  67. R. Wołak, "Graphs, Ehresmann connections, and vanishing cycles," In: Proc. Conf., Aug. 28-Sept. 1, 1995, Brno, Czech Republic, Brno (1996), pp. 345-352.

  68. R. Wołak, "On leaves of Lagrangian foliations," Russian Math., 6, 24-31 (1998).

    Google Scholar 

  69. N. I. Zhukova, "Graph of a foliation with Ehresmann connection and the stability of leaves," Izv. Vuzov, Mat., No. 2, 79-81 (1994).

    Google Scholar 

Download references

Authors
  1. M. A. Malakhal'tsev
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Malakhal'tsev, M.A. Foliations with Leaf Structures. Journal of Mathematical Sciences 108, 188–210 (2002). https://doi.org/10.1023/A:1012892219412

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1012892219412

Keywords

Search

Navigation