Skip to main content
Log in

Powers of the curvature operator of space forms and geodesics of the tangent bundle

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

It is well known that if Г is a geodesic line of the tangent (sphere) bundle with Sasaki metric of a locally symmetric Riemannian manifold, then all geodesic curvatures of the projected curve λ=π 1463-01 Г are constant. In this paper, we consider the case of the tangent (sphere) bundle over real, complex, and quaternionic space forms and give a unified proof of the following property: All geodesic curvatures of the projected curve are zero beginning with k 3, k 6, and k 10 for the real, complex, and quaternionic space forms, respectively.

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

Access this article

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. K. Sato, “Geodesics on the tangent bundles over space forms,” Tensor, 32, 5–10 (1978).

    Google Scholar 

  2. S. Sasaki, “Geodesics on the tangent sphere bundles over space forms,” J. Reine Angew. Math., 288, 106–120 (1976).

    Google Scholar 

  3. P. Nagy, “Geodesics on the tangent sphere bundle of a Riemannian manifold,” Geometria Didicata, 7, No.2, 233–244 (1978).

    Google Scholar 

  4. A. Yampol’skii, “Characterization of projections of geodesics of Sasaki metric of TCP n and T 1 CP n,” Ukr. Geom. Sb., 34, 121–126 (1991).

    Google Scholar 

  5. K. Azo, “A note on the projection curves of geodesics of the tangent and tangent sphere bundles,” Math. Repts Toyama Univ. (1988).

Download references

Author information

Authors and Affiliations

Authors

Additional information

__________

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 9, pp. 1231–1243, September, 2004.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sakharova, E., Yampol’skii, A. Powers of the curvature operator of space forms and geodesics of the tangent bundle. Ukr Math J 56, 1463–1480 (2004). https://doi.org/10.1007/s11253-005-0127-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-005-0127-5

Keywords

Navigation