Skip to main content
SpringerLink
Log in

Geometric interpretation of the Wagner curvature tensor in the case of a manifold with contact metric structure

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

Considering a manifold (φ, ξ, η, g, X, D) with contact metric structure, we introduce the concept of N-extended connection (connection on a vector bundle (D, π,X)), with N an endomorphism of the distribution D, and show that the curvature tensor of each N-extended connection for a suitably chosen endomorphism N coincides with the Wagner curvature tensor.

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. Wagner V. V., “Differential geometry of nonholonomic manifolds,” in: Reports: VIII International Lobachevsky Competition (1937), Kazan Fiz.-Mat. Obshch, Kazan, 1940.

    Google Scholar 

  2. Wagner V. V., “Geometry of an (n-1)-dimensional nonholonomic manifold in the n-dimensional space,” in: Tr. Semin. Vektorn. Tenzorn. Anal., Moscow Univ., Moscow, 1941, pp. 173–255.

    Google Scholar 

  3. Sasaki S., “On the differential geometry of tangent bundles of Riemannian manifolds,” Tohôku Math. J, No. 10, 338–354 (1958).

    Article  MathSciNet  MATH  Google Scholar 

  4. Bukusheva A. V. and Galaev S. V., “Almost contact metric structures defined by a connection over a distribution with admissible Finsler metric,” Izv. Saratovsk. Univ., Mat. Mekh. Inform., 12, No. 3, 17–22 (2012).

    MATH  Google Scholar 

  5. Bukusheva A. V. and Galaev S. V., “Connections on distributions and geodesic sprays,” Russian Math. (Iz. VUZ), 57, No. 4, 7–13 (2013).

    MathSciNet  MATH  Google Scholar 

  6. Galaev S. V., “The intrinsic geometry of almost contact metric manifolds,” Izv. Saratovsk. Univ., Mat. Mekh. Inform., 12, No. 1, 16–22 (2013).

    MATH  Google Scholar 

  7. Cartan E., “Sur les varietes à connexion affine et la theorie de la relative. I,” Ann. Sci. Éc. Norm. Super., 40, 325–412 (1923).

    MathSciNet  MATH  Google Scholar 

  8. Cartan E., “Sur les varietes à connexion affine et la theorie de la relative generalisee. I,” Ann. Sci. Éc. Norm. Super., 41, 1–25 (1924).

    MathSciNet  MATH  Google Scholar 

  9. Cartan E., “Sur les varietes à connexion affine et la theorie de la relative generalisee. II,” Ann. Sci. Éc. Norm. Super., 42, 17–88 (1925).

    MathSciNet  MATH  Google Scholar 

  10. Yano K., “On semi-symmetric metric connections,” Revue Roum. Math. Pures Appl., No. 15, 1579–1586 (1970).

    MATH  Google Scholar 

  11. Golab S., “On semi-symmetric and quarter-symmetric linear connections,” Tensor New Ser., 29, 249–254 (1975).

    MathSciNet  MATH  Google Scholar 

  12. Bejancu A., “Kähler contact distributions,” J. Geom. Phys., 60, 1958–1967 (2010).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Saratov State University, Saratov, Russia

    S. V. Galaev

Authors
  1. S. V. Galaev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. V. Galaev.

Additional information

Saratov. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 57, No. 3, pp. 632–640, May–June, 2016; DOI: 10.17377/smzh.2016.57.310.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Galaev, S.V. Geometric interpretation of the Wagner curvature tensor in the case of a manifold with contact metric structure. Sib Math J 57, 498–504 (2016). https://doi.org/10.1134/S0037446616030101

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0037446616030101

Keywords

Search

Navigation