Skip to main content
SpringerLink
Log in

Ehresmann connection for the canonical foliation on a manifold over a local algebra

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

For a canonical foliation on a manifoldM A over a local algebra, theA-affine horizontal distribution complementary to the leaves, similar to the horizontal distribution of a higher order connection on the fiber bundle ofA-jets, is defined. In the case of a complete manifoldM A, theA-affine horizontal distribution is proved to be an Ehresmann connection in the sense of Blumental-Hebda. It is shown that theA-affine horizontal distribution onM A exists if and only if the Atiyah class of a certain foliated principal bundle vanishes.

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

Similar content being viewed by others

References

  1. V. V. Shurygin, “Jet bundles as manifolds over algebras,” in:Itogi Nauki i Tekhniki, Problemy Geometrii [in Russian], Vol. 19, VINITI, Moscow (1987), pp. 3–22.

    Google Scholar 

  2. A. Weil, “Théorie des points proches sur les variétés différentiales,” in:Colloq. Internat. Centre Nat. Rech. Sci., Vol.52, Strasbourg, Paris (1953), p. 111–117.

    MATH  MathSciNet  Google Scholar 

  3. F. Brickell and R. S. Clark, “Integrable almost tangent structures,”J. Differential Geom.,9, No. 4, 557–563 (1974).

    MathSciNet  Google Scholar 

  4. O. Veblen and J. H. C. Whitehead,The Foundations of Differential Geometry, Cambridge (1932).

  5. V. V. Shurygin, “Manifolds over local algebras equivalent to jet bundles,”Izv. Vyssh. Uchebn. Zaved. Mat., [Soviet Math. J. (Iz. VUZ)], No. 10, 68–79 (1992).

    MATH  Google Scholar 

  6. R. A. Blumental and J. J. Hebda, “Ehresmann connections for foliations,”Indiana Math. J.,33, No. 4, 597–612 (1984).

    Google Scholar 

  7. R. A. Blumental and J. J. Hebda, “Complementary distributions which preserve the leaf geometry and applications to totally geodesic foliations,”Quart. J. Math.,35, 383–392 (1984).

    Google Scholar 

  8. P. Molino,Riemannian foliations, Birkhäuser, Boston-Basel (1988).

    Google Scholar 

  9. L. E. Evtushik, Yu. G. Lumiste, N. M. Ostianu, and A. P. Shirokov, “Differential geometry structures on manifolds,” in:Itogi Nauki i Tekhniki, Problemy Geometrii [in Russian], Vol. 9, VINITI, Moscow (1979).

    Google Scholar 

  10. S. Kobayasi and K. Nomizu,Foundations of Differential Geometry, Vol. 1, Interscience Publishers, New York-London (1981).

    Google Scholar 

  11. F. W. Kamber and P. Tondeur,Foliated Bundles and Characteristic Classes, Lect. Notes Math., Vol. 493, Springer-Verlag, Berlin-New York (1975).

    Google Scholar 

  12. B. N. Apanasov,Geometry of Discrete Manifolds [in Russian], Nauka, Moscow (1991).

    Google Scholar 

  13. D. Fried, W. Goldman, and M. W. Hirsch, “Affine manifolds with nilpotent holonomy,”Comment. Math. Helv. 56, No. 4, 487–523 (1981).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Kazan State University, USSR

    V. V. Shurygin

Authors
  1. V. V. Shurygin
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated fromMatematicheskie Zametki, Vol. 59, No. 2, pp. 303–310, February, 1996.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shurygin, V.V. Ehresmann connection for the canonical foliation on a manifold over a local algebra. Math Notes 59, 213–218 (1996). https://doi.org/10.1007/BF02310963

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02310963

Keywords

Search

Navigation