Skip to main content
SpringerLink
Log in

Isoparametric hypersurfaces in Finsler space forms

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we study isoparametric hypersurfaces in Finsler space forms by investigating focal points, tubes and parallel hypersurfaces of submanifolds. We prove that the focal submanifolds of isoparametric hypersurfaces are anisotropic-minimal and obtain a general Cartan-type formula in a Finsler space form with vanishing reversible torsion, from which we give some classifications on the number of distinct principal curvatures or their multiplicities.

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. Alves B, Javaloyes M A. A note on the existence of tubular neighbourhoods on Finsler manifolds and minimization of orthogonal geodesics to a submanifold. Proc Amer Math Soc, 2019, 147: 369–376

    Article  MathSciNet  Google Scholar 

  2. Bao D W, Chern S S, Shen Z M. An Introduction to Riemann-Finsler Geometry. Berlin-Heidelberg-New York: Springer, 2000

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Bryant R. Finsler structures on the 2-sphere satisfying K =1. In: Finsler Geometry. Contemporary Mathematics, vol. 193. Providence: Amer Math Soc, 1996, 27–42

    Chapter  Google Scholar 

  5. Cartan É. Familles de surfaces isoparamétriques dans les espaces à courbure constante. Ann Mat Pura Appl (4), 1938, 17: 177–191

    Article  Google Scholar 

  6. Cecil T E, Ryan P J. Geometry of Hypersurfaces. Springer Monographs in Mathematics. Berlin: Springer, 2015

    Book  Google Scholar 

  7. Chi Q S. Isoparametric hypersurfaces with four principal curvatures, IV. J Differential Geom, 2020, 115: 225–301

    Article  MathSciNet  Google Scholar 

  8. Dong P L, He Q. Isoparametric hypersurfaces of a class of Finsler manifolds induced by navigation problem in Minkowski spaces. Differential Geom Appl, 2020, 68: 101581

    Article  MathSciNet  Google Scholar 

  9. Ge J Q, Ma H. Anisotropic isoparametric hypersurfaces in Euclidean spaces. Ann Global Anal Geom, 2012, 41: 347–355

    Article  MathSciNet  Google Scholar 

  10. He Q, Yin S T, Shen Y B. Isoparametric hypersurfaces in Minkowski spaces. Differential Geom Appl, 2016, 47: 133–158

    Article  MathSciNet  Google Scholar 

  11. He Q, Yin S T, Shen Y B. Isoparametric hypersurfaces in Funk spaces. Sci China Math, 2017, 60: 2447–2464

    Article  MathSciNet  Google Scholar 

  12. Segre B. Famiglie di ipersuperficie isoparametrische negli spazi euclidei ad un qualunque numero di demesioni. Atti Accad Naz Lincei Cl Sci Fis Mat Natur Rend Lincei (9) Mat Appl, 1938, 27: 203–207

    MATH  Google Scholar 

  13. Shen Y B, Shen Z M. Introduction to Modern Finsler Geometry. Beijing: Higher Education Press, 2016

    Book  Google Scholar 

  14. Shen Z M. Lectures on Finsler Geometry. Singapore: World Scientific, 2001

    Book  Google Scholar 

  15. Somigliana C. Sulle relazione fra il principio di Huygens e l’ottica geometrica. Atti Accad Sci Torino Cl Sci Fis Mat Natur, 1918, 24: 974–979

    MATH  Google Scholar 

  16. Xu M. Isoparametric hypersurfaces in a Randers sphere of constant flag curvature. Ann Mat Pura Appl (4), 2018, 197: 703–720

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11971253 and 11471246) and AnHui Natural Science Foundation (Grant No. 1608085MA03).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Tongji University, Shanghai, 200092, China

    Qun He, Yali Chen & Tingting Ren

  2. Department of Mathematics and Computer Science, Tongling University, Tongling, 244000, China

    Songting Yin

Authors
  1. Qun He
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yali Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Songting Yin
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Tingting Ren
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Songting Yin.

Additional information

In Memory of Professor Zhengguo Bai (1916–2015)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

He, Q., Chen, Y., Yin, S. et al. Isoparametric hypersurfaces in Finsler space forms. Sci. China Math. 64, 1463–1478 (2021). https://doi.org/10.1007/s11425-020-1804-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-020-1804-6

Keywords

MSC(2010)

Search

Navigation