Skip to main content
SpringerLink
Log in

On Ricci tensor of focal submanifolds of isoparametric hypersurfaces

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

Abstract

A-manifolds and B-manifolds, introduced by Gray (1978), are two significant classes of Einstein-like Riemannian manifolds. A Riemannian manifold is Ricci parallel if and only if it is simultaneously an A-manifold and a B-manifold. The present paper proves that both focal submanifolds of each isoparametric hypersurface in unit spheres with g = 4 distinct principal curvatures are A-manifolds. As for the focal submanifolds with g = 6, m = 1 or 2, only one is an A-manifold, and neither is a B-manifold.

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. Abresch U. Isoparametric hypersurfaces with four or six distinct principal curvatures. Math Ann, 1983, 264: 283–302

    Article  MathSciNet  Google Scholar 

  2. Cartan E. Sur des familles remarquables d’hypersurfaces isoparamétriques dans les espaces sphériques. Math Z, 1939, 45: 335–367

    Article  MathSciNet  Google Scholar 

  3. Cecil T E, Chi Q S, Jensen G R. Isoparametric hypersurfaces with four principal curvatures. Ann Math, 2007, 166: 1–76

    Article  MathSciNet  Google Scholar 

  4. Cecil T E, Ryan P T. Tight and Taut Immersions of Manifolds. Research Notes in Math, vol. 107. London: Pitman, 1985

    Google Scholar 

  5. Chi Q S. Isoparametric hypersurfaces with four principal curvatures, III. J Diff Geom, 2013, 94: 487–522

    Google Scholar 

  6. Dorfmeister J, Neher E. Isoparametric hypersurfaces, case g = 6, m = 1. Comm Algebra, 1985, 13: 2299–2368

    Article  MathSciNet  Google Scholar 

  7. Ferus D, Karcher H, Münzner H F. Cliffordalgebren und neue isoparametrische Hyperflächen. Math Z, 1981, 177: 479–502

    Article  MathSciNet  Google Scholar 

  8. Ge J Q, Qian C. Finiteness results for equifocal hypersurfaces in compact symmetric spaces. Sci China Math, 2014, 57: 1975–1982

    Article  MathSciNet  Google Scholar 

  9. Ge J Q, Tang Z Z. Isoparametric functions and exotic spheres. J Reine Angew Math, 2013, 683: 161–180

    MathSciNet  Google Scholar 

  10. Gray A. Einstein like manifolds which are not Einstein. Geom Ded, 1978, 7: 259–280

    Article  Google Scholar 

  11. Hsiang W Y, Lawson H B. Minimal submanifolds of low cohomogenity. J Diff Geom, 1971, 5: 1–38

    MathSciNet  Google Scholar 

  12. Immervoll S. On the classification of isoparametric hypersurfaces with four distinct principal curvatures in spheres. Ann Math, 2008, 168: 1011–1024

    Article  MathSciNet  Google Scholar 

  13. Miyaoka R. The linear isotropy group of G 2 /SO(4), the Hopf fibering and isoparametric hypersurfaces. Osaka J Math, 1993, 30: 179–202

    MathSciNet  Google Scholar 

  14. Miyaoka R. Geometry of G 2 orbits and isoparametric hypersurfaces. Nagoya Math J, 2011, 203: 175–189

    Article  MathSciNet  Google Scholar 

  15. Miyaoka R. Isoparametric hypersurfaces with (g,m) = (6, 2). Ann Math, 2013, 177: 53–110

    Article  MathSciNet  Google Scholar 

  16. Münzner H F. Isoparametrische Hyperflächen in Sphären, I and II. Math Ann, 1980, 251: 57–71; 1981, 256: 215–232

    Article  MathSciNet  Google Scholar 

  17. Ozeki H, Takeuchi M. On some types of isoparametric hypersurfaces in spheres, I and II. Tôhoku Math J, 1975, 27: 515–559; 1976, 28: 7–55

    Article  Google Scholar 

  18. Qian C, Tang Z Z, Yan W J. New examples of Willmore submanifolds in the unit sphere via isoparametric functions, II. Ann Glob Anal Geom, 2013, 43: 47–62

    Article  MathSciNet  Google Scholar 

  19. Tang Z Z, Xie Y Q, Yan W J. Schoen-Yau-Gromov-Lawson theory and isoparametric foliation. Comm Anal Geom, 2012, 20: 989–1018

    Article  MathSciNet  Google Scholar 

  20. Tang Z Z, Yan W J. New examples of Willmore submanifolds in the unit sphere via isoparametric functions. Ann Glob Anal Geom, 2012, 42: 403–410

    Article  MathSciNet  Google Scholar 

  21. Tang Z Z, Yan W J. Isoparametric foliation and a problem of Besse on generalizations of Einstein condition. ArXiv:1307.3807, 2013

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing, 100875, China

    QiChao Li & WenJiao Yan

Authors
  1. QiChao Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. WenJiao Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to WenJiao Yan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, Q., Yan, W. On Ricci tensor of focal submanifolds of isoparametric hypersurfaces. Sci. China Math. 58, 1723–1736 (2015). https://doi.org/10.1007/s11425-014-4928-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-014-4928-6

Keywords

MSC(2010)

Search

Navigation