Skip to main content
SpringerLink
Log in

Distinguished Connections on Finsler Algebroids

  • Published:
Chinese Annals of Mathematics, Series B Aims and scope Submit manuscript

Abstract

Considering the prolongation of a Lie algebroid, the authors introduce Finsler algebroids and present important geometric objects on these spaces. Important endomorphisms like conservative and Barthel, Cartan tensor and some distinguished connections like Berwald, Cartan, Chern-Rund and Hashiguchi are introduced and studied.

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. Bao, D., Chern, S. and Shen, Z., An Introduction to Riemann-Finsler Geometry, Springer-Verlag, New York, 2000.

    Book  Google Scholar 

  2. Cortez, J., de Leon, M., Marrero, J. C. et al., A survey of Lagrangian mechanics and control on Lie algebroids and groupoids, Int. J. Geom. Methods Mod. Phys., 3(3), 2006, 509–558.

    Article  MathSciNet  Google Scholar 

  3. Grabowski, J. and Urbański, P., Tangent and cotangent lift and graded Lie algebra associated with Lie algebroids, Ann. Global Anal. Geom., 15, 1997, 447–486.

    Article  MathSciNet  Google Scholar 

  4. Grabowski, J. and Urbański, P., Lie algebroids and Poisson-Nijenhuis structures, Rep. Math. Phys., 40, 1997, 195–208.

    Article  MathSciNet  Google Scholar 

  5. de León, M., Marrero, J. C. and Martínez, E., Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A: Math. Gen., 38, 2005, 241–308.

    Article  MathSciNet  Google Scholar 

  6. Martinez, E., Lagrangian mechanics on Lie algebroids, Acta Appl. Math., 67, 2001, 295–320.

    Article  MathSciNet  Google Scholar 

  7. Mestdag, T., A Lie algebroid approach to Lagrangian systems with symmetry, Diff. Geom. Appl., 2005, 523–535.

  8. Peyghan, E., Berwald-type and Yano-type connections on Lie algebroids, Int. J. Geom. Methods Mod. Phys., 12 (10), 2015, 36 pages.

  9. Popescu, L., The geometry of Lie algebroids and applications to optimal control, Annals. Univ. Al. I. Cuza, Iasi, Series I, Math., LI, 2005, 155–170.

    MathSciNet  MATH  Google Scholar 

  10. Popescu, L., Aspects of Lie algebroids geometry and Hamiltonian formalism, Annals Univ. Al. I. Cuza, Iasi, Series I, Math., LIII(supl.), 2007, 297–308.

    MathSciNet  MATH  Google Scholar 

  11. Popescu, L., A note on Poisson-Lie algebroids, J. Geom. Symmetry Phys., 12, 2008, 63–73.

    MathSciNet  MATH  Google Scholar 

  12. Pradines, J., Théorie de Lie pour les groupoïdes différentiables. Calcul différentiel dans la catégorie des groupoïdes infinitésimaux, C. R. Acad. Sci. Paris, 264(A), 1967, 245–248.

    MathSciNet  MATH  Google Scholar 

  13. Szakál, Sz. and Szilasi, J., A new approach to generalized Berwald manifolds I, SUT J. Math., 37, 2001, 19–41.

    MathSciNet  MATH  Google Scholar 

  14. Szakál, Sz. and Szilasi, J., A new approach to generalized Berwald manifolds I, Publ. Math. Debrecen, 60, 2002, 429–453.

    MathSciNet  MATH  Google Scholar 

  15. Szilasi, J., A setting for spray and finsler geometry, Handbook of Finsler Geometry, Kluwer Academic Publishers, Dordrecht, 2003, 1183–1426.

    MATH  Google Scholar 

  16. Szilasi, J. and Győry, A., A generalization of Weyl’s theorem on projectively related affine connections, Rep. Math. Phys., 54, 2004, 261–273.

    Article  MathSciNet  Google Scholar 

  17. Szilasi, J. and Tóth, A., Conformal vector fields on Finsler manifolds, Comm. Math., 19, 2011, 149–168.

    MathSciNet  MATH  Google Scholar 

  18. Vacaru, S. I., Nonholonomic algebroids, Finsler geometry, and Lagrange-Hamilton spaces, Mathematical Sciences, 6, 2012, 1–33.

    Article  MathSciNet  Google Scholar 

  19. Vacaru, S. I., Almost Kähler Ricci flows and Einstein and Lagrange-Finsler structures on Lie algebroids, 2013, arXiv: submit/0724182 [math-ph].

  20. Weinstein, A., Lagrangian mechanics and grupoids, Fields Institute Communications, 7, 1996, 207–231.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Arak University, Arak, 38156-8-8349, Iran

    Esmaeil Peyghan

  2. Faculty of Science, Department of Mathematics, Ataturk University, 25240, Erzurum, Turkey

    Aydin Gezer & Inci Gultekin

Authors
  1. Esmaeil Peyghan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Aydin Gezer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Inci Gultekin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Aydin Gezer.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Peyghan, E., Gezer, A. & Gultekin, I. Distinguished Connections on Finsler Algebroids. Chin. Ann. Math. Ser. B 42, 41–68 (2021). https://doi.org/10.1007/s11401-021-0244-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11401-021-0244-y

Keywords

2000 MR Subject Classification

Search

Navigation