Skip to main content
SpringerLink
Log in

Left-invariant metrics on a tensor bundle of type (2,0) over a Lie group

  • Published:
Lobachevskii Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper, vertical and horizontal lifts of left-invariant vector fields are constructed. Necessary and sufficient conditions for the horizontal lift of left-invariant vector fields to be left-invariant field are established. On the basis of a left-invariant metric on G, left-invariant vertical and horizontal distributions and a left-invariant metric g on T 20 G are constructed.

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. B. N. Shapukov, “Tensor bundles,” in Dedicated to the Memory of Lobachevskii, Ed. by A.P. Shirokov (Izd. Kazan. Univ., Kazan, 1992), Vol. 1, pp. 104–125.

    Google Scholar 

  2. N. A. Opokina, “Tangent and tensor bundles of (2, 0) type under Lie group,” Uch. Zap., Kazan. Gos. Univ., Ser. Fiz.-Mat. Nauki, 147(1), 138–147 (2005).

    MATH  Google Scholar 

  3. M. M. Postnikov, Lectures in Geometry. Semester 4: Differential Geometry (Nauka, Moscow, 1988) [in Russian].

    Google Scholar 

  4. B. N. Shapukov, Problems on Lie Groups and their Applications (RKhD, Moscow, 2002) [in Russian].

    Google Scholar 

  5. N. A. Opokina, “Left connection on a tensor bundle of type (2, 0) over a Lie group,” Russian Math. (Iz. VUZ) 50(11), 74–79 (2006).

    MathSciNet  Google Scholar 

  6. L. P. Eisenhart, Continuous Groups of Transformations (Dover, New York, 1933; Editorial URRS, Moscow, 2004).

    MATH  Google Scholar 

  7. Sh. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. 1 (Interscience, New York, 1969; Nauka, Moscow, 1987).

    MATH  Google Scholar 

  8. M. M. Postnikov, Lectures in Geometry. Semester 5: Riemannian Geometry (Faktorial, Moscow, 1998) [in Russian].

    Google Scholar 

  9. S. P. Gavrilov, “Geodesics of left-invariant metrics on a connected two-dimensional nonabelian Lie group,” in Gravitation and Relativity Theory, Ed. by V. R. Kaigorodov (Izd. Kazan. Univ., Kazan’ 1981), Vol. 18, pp. 28–44 [in Russian].

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Kazan Federal University, Kremlevskaya ul. 18, Kazan, 420008, Tatarstan, Russia

    N. A. Opokina

Authors
  1. N. A. Opokina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. A. Opokina.

Additional information

Original Russian Text © N.A. Opokina, 2012, published in Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2012, Vol. 154, No. 4, pp. 146–155.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Opokina, N.A. Left-invariant metrics on a tensor bundle of type (2,0) over a Lie group. Lobachevskii J Math 34, 384–391 (2013). https://doi.org/10.1134/S1995080213040045

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1995080213040045

Keywords and phrases

Search

Navigation