Metrics and Connections on the Bundle of Affinor Frames


In this paper the authors consider the bundle of affinor frames over a smooth manifold, define the Sasaki metric on this bundle, and investigate the Levi-Civita connection of Sasaki metric. Also the authors determine the horizontal lifts of symmetric linear connection from a manifold to the bundle of affinor frames and study the geodesic curves corresponding to the horizontal lift of the linear connection.

This is a preview of subscription content, access via your institution.


  1. [1]

    Aso, K., Notes on some properties of the sectional curvature of the tangent bundle, Yakohama Math. J., 29(3), 1981, 1–5.

    MathSciNet  MATH  Google Scholar 

  2. [2]

    Cordero, L. A. and de Leon, M., Horizontal lift of connection to the frame bundle, Boll. Un. Mat. Ital., 6, 1984, 223–240.

    MathSciNet  MATH  Google Scholar 

  3. [3]

    Fattayev, H. D., Transfer of some differential geometric structures from (1, 1)- tensor bundle to the (0, 2)-tensor bundle over a Riemannian manifold, Trans. of NAS of Azerbaijan, Issue Math., Ser. of Phys. Tech. and Math. Sciences, 38(1), 2018, 52–61.

    MathSciNet  MATH  Google Scholar 

  4. [4]

    Fattayev, H. D., Some problems of differential geometry of the bundle of affinor frames, News of Baku Univ., Ser. of Physico-Math. Sciences, (3), 2018, 45–57.

  5. [5]

    Fattayev, H. D. and Salimov, A. A., Diagonal lifts of metrics to coframe bundle, Proc. of IMM of NAS of Azerbaijan, 44(2), 2018, 328–337.

    MathSciNet  MATH  Google Scholar 

  6. [6]

    Kowalski, O., Curvature of the induced Riemannian metric on the tangent bundle of Riemannian manifold, J. Reine Angew. Math., 250, 1971, 124–129.

    MathSciNet  MATH  Google Scholar 

  7. [7]

    Kowalski, O. and Sekizawa, M., On curvatures of linear frame bundle with naturally lifted metrics, Rend. Sem. Mat. Univ. Pol. Torino, 63, 2005, 283–296.

    MathSciNet  MATH  Google Scholar 

  8. [8]

    Kurek, J., On a horizontal lift of a linear connection to the bundle of linear frames, Ann. Univ. Marie Curie-Skladowska. Sectio A, XLI, 1987, 31–37.

    MathSciNet  MATH  Google Scholar 

  9. [9]

    Mok, K. P., Metrics and connections on the cotangent bundle, Kodai Math. Sem. Rep., 28, 1977, 226–238.

    MathSciNet  Article  Google Scholar 

  10. [10]

    Mok, K. P., On the differential geometry of frame bundles of Riemannian manifolds, J. Reine Angew. Math., 302, 1978, 16–31.

    MathSciNet  MATH  Google Scholar 

  11. [11]

    Musso, E. and Tricerri, F., Riemannian metrics on tangent bundles, Ann. Mat. Pura Appl., 150, 1988, 1–20.

    MathSciNet  Article  Google Scholar 

  12. [12]

    Salimov, A. A. and Cengiz, N., Lifting of Riemannian metrics to tensor bundles, Russian Math. (Iz. Vuz.), 47(11), 2003, 47–55.

    MathSciNet  MATH  Google Scholar 

  13. [13]

    Salimov, A. A. and Agca, F., Some properties of Sasakian metrics in cotangent bundles, Mediterr. J. Math., 8, 2011, 243–255.

    MathSciNet  Article  Google Scholar 

  14. [14]

    Salimov, A. A. and Gezer, A., On the geometry of the (1, 1)-tensor bundle with Sasaki type metric, Chin. Ann. Math., 32, 2011, 1–18.

    MathSciNet  Article  Google Scholar 

  15. [15]

    Salimov, A. A., Gezer, A. and Akbulut, K., Geodesics of Sasakian metrics on tensor bundles, Mediterr. J. Math., 6(2), 2009, 137–149.

    MathSciNet  Article  Google Scholar 

  16. [16]

    Sasaki, S., On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J., 10, 1958, 338–354.

    MathSciNet  Article  Google Scholar 

  17. [17]

    Yano, K. and Ishihara, S., Horizontal lifts of tensor fields and connections to tangent bundles, Math. and Mech., 16, 1967, 1015–1030.

    MathSciNet  MATH  Google Scholar 

  18. [18]

    Yano, K. and Ishihara, S., Tangent and Cotangent Bundles, Marsel Dekker, Inc., New York, 1973.

    Google Scholar 

  19. [19]

    Yano, K. and Patterson, E. M., Horizontal lift from a manifold to its cotangent bundle, J. Math. Soc. Japan, 19, 1967, 185–198.

    MathSciNet  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Arif Salimov.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Fattayev, H., Salimov, A. Metrics and Connections on the Bundle of Affinor Frames. Chin. Ann. Math. Ser. B 42, 121–134 (2021).

Download citation


  • Bundle of affinor frames
  • Riemannian manifold
  • Sasaki metric
  • Horizontal lift
  • Geodesic curve

2000 MR Subject Classification

  • 53C05
  • 53B20