Skip to main content
Log in

Recent progress on homogeneous Finsler spaces with positive curvature

  • Review Article
  • Published:
European Journal of Mathematics Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

We survey our recent works concerning the classification of positively curved homogeneous Finsler spaces, and some related topics. In the final part, we present some open problems in this field.

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.

Similar content being viewed by others

References

  1. Aloff, S., Wallach, N.R.: An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures. Bull. Amer. Math. Soc. 81(1), 93–97 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  2. Álvarez Paiva, J.C., Durán, C.E.: Isometric submersions of Finsler manifolds. Proc. Amer. Math. Soc. 129(8), 2409–2417 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bao, D., Chern, S.-S., Shen, Z.: An Introduction to Riemann–Finsler Geometry. Graduate Texts in Mathematics, vol. 200. Springer, New York (2000)

    Book  MATH  Google Scholar 

  4. Bao, D., Robles, C., Shen, Z.: Zermelo navigation on Riemannian manifolds. J. Differential Geom. 66(3), 377–435 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bérard-Bergery, L.: Les variétés riemanniennes homogénes simplement connexes de dimension impaire à courbure strictement positive. J. Math. Pures Appl. 55(1), 47–67 (1976)

    MathSciNet  MATH  Google Scholar 

  6. Berger, M.: Les variétés riemanniennes homogènes normales simplement connexes à courbure strictement positive. Ann. Sc. Norm. Super. Pisa 15, 179–246 (1961)

    MATH  Google Scholar 

  7. Borel, A.: Some remarks about Lie groups transitive on spheres and tori. Bull. Amer. Math. Soc. 55(6), 580–587 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chern, S.-S., Shen, Z.: Riemann–Finsler Geometry. Nankai Tracts in Mathematics, vol. 6. World Scientific, Hackensack (2005)

    MATH  Google Scholar 

  9. Deng, S.: Fixed points of isometries of a Finsler space. Publ. Math. Debrecen 72(3–4), 469–474 (2008)

    MathSciNet  MATH  Google Scholar 

  10. Deng, S.: Invariant Finsler metrics on polar homogeneous spaces. Pacific J. Math. 247(1), 47–74 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Deng, S.: Homogeneous Finsler Spaces. Springer Monographs in Mathematics. Springer, New York (2012)

    Book  MATH  Google Scholar 

  12. Deng, S., Hou, Z.: Homogeneous Finsler spaces of negative curvature. J. Geom. Phys. 57(2), 657–664 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Deng, S., Hou, Z.: Homogeneous Einstein–Randers spaces of negative Ricci curvature. C. R. Math. Acad. Sci. Paris 347(19–20), 1169–1172 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Deng, S., Hu, Z.: Curvatures of homogeneous Randers spaces. Adv. Math. 240, 194–226 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hu, Z., Deng, S.: Homogeneous Randers spaces with positive flag curvature and isotropic S-curvature. Math. Z. 270(3–4), 989–1009 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Huang, L.: On the fundamental equations of homogeneous Finsler spaces. Differential Geom. Appl. 40, 187–208 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kobayashi, S.: Homogeneous Riemannian manifolds of negative curvature. Tôhoku Math. J. 14(4), 413–415 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mo, X., Hang, L.: On curvature decreasing property of a class of navigation problems. Publ. Math. Debrecen 71(1–2), 141–163 (2007)

    MathSciNet  MATH  Google Scholar 

  19. Montgomery, D., Samelson, H.: Transformation groups of spheres. Ann. Math. 44(3), 454–470 (1943)

    Article  MathSciNet  MATH  Google Scholar 

  20. Shen, Z.: Volume comparison and its applications in Riemann–Finsler geometry. Adv. Math. 128(2), 306–328 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  21. Shen, Z.: Lectures on Finsler Geometry. World Scientific, Singapore (2001)

    Book  MATH  Google Scholar 

  22. Shen, Z.: Differential Geometry of Spray and Finsler Spaces. Kluwer Academic, Dordrecht (2001)

    Book  MATH  Google Scholar 

  23. Verdiani, L., Ziller, W.: Positively curved homogeneous metrics on spheres. Math. Z. 261(3), 473–488 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wallach, N.R.: Compact homogeneous Riemannian manifolds with strictly positive curvature. Ann. Math. 96(2), 277–295 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wilking, B.: The normal homogeneous space \(({\rm SU}(3){\times }{\rm SO}(3))/{\rm U}^\bullet (2)\) has positive sectional curvature. Proc. Amer. Math. Soc. 127(4), 1191–1194 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. Wilking, B.: Positively curved manifolds with symmetry. Ann. Math. 163(2), 607–668 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wilking, B., Ziller, W.: Revisiting homogeneous spaces with positive curvature. J. Reine Angew. Math. doi:10.1515/crelle-2015-0053. arXiv:1503.06256

  28. Xu, M.: Examples of flag-wise positively curved spaces. Differential Geom. Appl. 52, 42–50 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  29. Xu, M., Deng, S.: Homogeneous \((\alpha,\beta )\)-spaces with positive flag curvature and vanishing S-curvature. Nonlinear Anal. 127, 45–54 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Xu, M., Deng, S.: Normal homogeneous Finsler spaces. Transform. Groups. arXiv:1411.3053

  31. Xu, M., Deng, S.: Towards the classification of odd-dimensional homogeneous reversible Finsler spaces with positive flag curvature. Ann. Mat. Pura Appl. doi:10.1007/s10231-016-0624-1. arXiv:1504.03018

  32. Xu, M., Deng, S.: Homogeneous Finsler spaces and the flag-wise positively curved condition (2016). arXiv:1604.07695

  33. Xu, M., Deng, S., Huang, L., Hu, Z.: Even dimensional homogeneous Finsler spaces with positive flag curvature. Indiana Univ. Math. J. arXiv:1407.3582

  34. Xu, M., Wolf, J.A.: \({\rm Sp} (2)/\rm U(1)\) and a positive curvature problem. Differential Geom. Appl. 42, 115–124 (2015)

    Article  MathSciNet  Google Scholar 

  35. Xu, M., Zhang, L.: \(\delta \)-homogeneity in Finsler geometry and the positive curvature problem. Osaka J. Math. arXiv:1611.00920

  36. Xu, M., Ziller, W.: Reversible homogeneous Finsler metrics with positive flag curvature. Forum Math. doi:10.1515/forum-2016-0173. arXiv:1606.02474

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ming Xu.

Additional information

Supported by NSFC (nos. 11671212, 51535008), SRFDP of China.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Deng, S., Xu, M. Recent progress on homogeneous Finsler spaces with positive curvature. European Journal of Mathematics 3, 974–999 (2017). https://doi.org/10.1007/s40879-017-0148-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40879-017-0148-2

Keywords

Mathematics Subject Classification

Navigation