Advertisement

Geometriae Dedicata

, Volume 171, Issue 1, pp 119–148 | Cite as

Kropina metrics and Zermelo navigation on Riemannian manifolds

  • Ryozo Yoshikawa
  • Sorin V. SabauEmail author
Original Paper

Abstract

The present paper studies globally defined Kropina metrics as solutions of the Zermelo’s navigation problem. Moreover, we characterize the Kropina metrics of constant flag curvature showing that up to local isometry, there are only two model spaces of them: the Euclidean space and the odd-dimensional spheres.

Keywords

Kropina spaces Flag curvature Killing vector fields Riemannian isometries 

Mathematics Subject Classification (2000)

53C60 53C20 

Notes

Acknowledgments

We thank to Professors K. Okubo and H. Shimada for encouraging us in the research of this topic.

References

  1. 1.
    Antonelli, P.L., Ingarden, R.S., Matsumoto, M.: The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology. Kluwer, Dordrecht (1993)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bao, D., Chern, S.S., Shen, Z.: An Introduction to Riemann Finsler Geometry. Springer, GTM 200, (2000)Google Scholar
  3. 3.
    Bao, D., Robles, C., Shen, Z.: Zermelo navigation on Riemannian manifolds. J. Differ. Geom. 66, 377–435 (2004)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Berestovskii, V.N., Nikonorov, Y.G.: Killing vector field of constant length on Riemannian manifolds. Sib. Math. J. 49(3), 395407 (2008)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Berger, M.: Trois remarques sur les variétés riemanniennes \(\grave{a}\) courbure positive. C. R. Acad. Sci. Paris Sér. A-B 263, 76–78 (1966)zbMATHGoogle Scholar
  6. 6.
    Bernestovskii, V.N., Nikonorov, Y.G.: Clifford-Wolf homogeneous Riemannian manifold. J. Differ. Geom. 82, 467–500 (2009)Google Scholar
  7. 7.
    Carathéodory, C.: Calculus of Variations and Partial Differential Equations of the First Order, (Translated by Robert B. Dean). AMS Chelsea Publishing, Berlin (2006). (Originally published 1935, Berlin)Google Scholar
  8. 8.
    DeVito, J: Curvature of Invariant Metrics on Compact Lie Groups, Web site: www.math.upenn.edu/devito/curv.pdf, (2003)
  9. 9.
    Do Carmo, M.P.: Riemannian Geometry. Birkhäuser, Boston (1993)Google Scholar
  10. 10.
    Javaloyes, M.A., Sánchez, M.: On the definition and examples of Finsler metrics, arXiv. 1111.5066v1 [math. DG] (2011)Google Scholar
  11. 11.
    Kleiner, B., Lott, J.: Notes on Perelman’s papers. Geom. Topol. 12(5), 25872855 (2008)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Kropina, V.K.: On projective two-dimensional finsler spaces with special metric. Trudy Sem. Vector. Tensor. Anal. 11, 277 (1961)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Matsumoto, M.: Projectively flat Finsler spaces with \((\alpha, \beta )-\)metric. Rep. Math. Phys. 30, 15–20 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Matsumoto, M.: Finsler spaces of constant curvature with Kropina metric. Tensor N. S. 50, 194–201 (1991)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Robles, C.: Geodesics in randers spaces of constant curvature. Trans. Am. Math. Soc. 359(4), 1633–1651 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Shen, Z.: Finsler metric with \(K=0\) and \(S=0\). Can. J. Math. 55, 112–132 (2003)CrossRefzbMATHGoogle Scholar
  17. 17.
    Shibata, C.: On Finsler spaces with Kropina metric. Rep. Math. Phys. 13, 117–128 (1978)CrossRefzbMATHGoogle Scholar
  18. 18.
    Yoshikawa, R., Okubo, K.: Kropina spaces of constant curvature. Tensor N.S. 68, 190–203 (2007)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Yoshikawa, R., Okubo, K.: Constant curvature conditions for Kropina spaces. Balkan J. Geom. Appl. 17(1), 115–124 (2012)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Hachiman Technical High SchoolHachimanJapan
  2. 2.Department of MathematicsTokai UniversitySapporoJapan

Personalised recommendations