Advertisement

Results in Mathematics

, 74:147 | Cite as

Equiaffine Isoparametric Functions and Their Regular Level Hypersurfaces

  • Wenjing Hao
  • Xingxiao LiEmail author
Article
First Online:
  • 62 Downloads

Abstract

In this paper, we aim to introduce and study the (locally strongly convex) equiaffine isoparametric functions on the affine space \(A^{n+1}\), making the emphasis on their relation with the affine isoparametric hypersurfaces. Motivated by the case in the Euclidean space \(E^{n+1}\), we first introduce the concept of equiaffine parallel hypersurfaces in \(A^{n+1}\), and then equivalently re-define the equiaffine isoparametric hypersurfaces to be ones that are among families of equiaffine parallel hypersurfaces in \(A^{n+1}\) of constant affine mean curvature. As the main result, we prove that an equiaffine isoparametric hypersurface is nothing but exactly a regular level set of some equiaffine isoparametric function.

Keywords

Affine isoparametric function affine isoparametric hypersurfaces affine parallel hypersurfaces affine principal curvature affine mean curvature 

Mathematics Subject Classification

Primary 53A15 Secondary 53B25 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

The second author thanks Professor A.-M. Li for his constant encouragement.

References

  1. 1.
    Cartan, E.: Families des surface isoparametrique dans les espaces a courbure constante. Ann. Mat. 17, 177–191 (1938)CrossRefGoogle Scholar
  2. 2.
    Cartan, E.: Sur des familles remarquables dhypersurfaces isoparamtriques dans les espaces sphriques. Math. Z. 45, 335–367 (1939)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cecil, T.E.: Isoparametric and Dupin hypersurfaces. Symmetry Integr. Geom. Methods Appl. (SIGMA) 4, 062 (2008).  https://doi.org/10.3842/SIGMA.2008.062 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cecil, T.E., Chi, Q.S., Jensen, G.R.: Isoparametric hypersurfaces with four principal curvatures. Ann. Math. 166, 1–76 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cecil, T.E., Ryan, P.T.: Tight and Taut Immersions of Manifolds, Research Notes in Mathematics, vol. 107. Pitman, London (1985)zbMATHGoogle Scholar
  6. 6.
    Chi, Q.S.: The Isoparametric Story. National Taiwan University, Taipei City (2012)Google Scholar
  7. 7.
    Ge, J.Q., Tang, Z.Z.: Isoparametric functions and exotic spheres. J. Reine Angew. Math. 683, 161–180 (2013)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Ge, J.Q., Tang, Z.Z.: Geometry of isoparametric hypersurfaces in Riemannian manifolds. Asian J. Math. 18(1), 117–126 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ge, J.Q., Xie, Y.Q.: Gradient map of isoparametric polynomial and its application to Ginzburg-Landau system. J. Funct. Anal. 258, 1682–1691 (2010)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hu, Z.J., Li, X.X., Jie, S.J.: On the Blaschke isoparametric hypersurfaces in the unit sphere with three distinct Blaschke eigenvalues. Sci. China Math. 54(10), 2171–2194 (2011).  https://doi.org/10.1007/s11425-011-4291-9 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Immervoll, S.: On the classification of isoparametric hypersurfaces with four distinct principal curvatures in spheres. Ann. Math. 168, 1011–1024 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Koike, N.: Blaschke Dupin hypersurfaces and equiaffine tubes. Results Math. 48, 97–108 (2005)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Li, A.-M., Simon, U., Zhao, G.S.: Global Affine Differential Geometry of Hypersurfaces. Walter de Gruyter and Co, Berlin (1993)CrossRefGoogle Scholar
  14. 14.
    Li, H.Z., Liu, H.L., Wang, C.P., Zhao, G.S.: Möbius isoparametric hypersurfaces in \(S^{n+1}\) with two distinct principal curvatures. Acta Math. Sin. (Eng. Ser.) 18, 437–446 (2002)CrossRefGoogle Scholar
  15. 15.
    Li, T.Z., Qing, J., Wang, C.P.: Möbius and Laguerre Geometry of Dupin Hypersurfaces. arXiv:1503.02914v1 [math.DG]
  16. 16.
    Li, T.Z., Wang, C.P.: A note on Blaschke isoparametric hypersurfaces. Int. J. Math. 25, 1450117 (2014).  https://doi.org/10.1142/S0129167X14501171 MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Li, X.X., Zhang, F.Y.: Immersed hypersurfaces in the unit sphere \(S^{m+1}\) with constant Blaschke eigenvalues. Acta Math. Sin. Engl. Ser. 23, 533–548 (2007)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Li, X.X., Zhang, F.Y.: On the Blaschke isoparametric hypersurfaces in the unit sphere. Acta Math. Sin. Engl. Ser. 25, 657–678 (2009)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Münzner, H.F.: Isoparametric hyperflächen in sphären, I. Math. Ann. 251, 57–71 (1980)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Münzner, H.F.: Isoparametric hyperflächen in sphären, II. Math. Ann. 256, 215–232 (1981)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Niebergall, R., Ryan, P.J.: Isoparametric Hypersurfaces-the Affine Case, Geometry and Topology of Submanifolds, V, vol. 201-214. World Scientific Publishing, River Edge (1993)zbMATHGoogle Scholar
  22. 22.
    Niebergall, R., Ryan, P.J.: Focal Sets in Affine Geometry, Geometry and Topology of Submanifolds VI. World Scientific, River Edge (1994)zbMATHGoogle Scholar
  23. 23.
    Niebergall, R., Ryan, P.J.: Affine isoparametric hypersurfaces. Math. Z. 217, 479–485 (1994)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Nomizu, K., Sasaki, T.: Affine Differential Geometry. Cambridge University Press, Cambridge (1994)zbMATHGoogle Scholar
  25. 25.
    Ooguri, M.: On Carlan’s identities of equiaffine isoparametric hypersurfaces. Results Math. 46, 79–90 (2004)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Rodrigues, L.A., Tenenblat, K.: A characterization of Moebius isoparametric hypersurfaces of the sphere. Monatsh. Math. 158, 321–327 (2009)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Stamatakis, S., Kaffas, I., Delivos, I.: Generalization of two Bonnets Theorems to the relative differential geometry of the \(3\)-dimensional Euclidean space. J. Geom. 108, 1073–1082 (2017)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Stamatakis, S., Kaffas, I.: Bonnet’s Type Theorems in the Relative Differential Geometry of the \(4\)-Dimensional Space. arXiv:1707.07549 [math.DG] (2017)
  29. 29.
    Stamatakis, S., Kaffas, I.: On the Shape Operator of Relatively Parallel Hypersurfaces in the \(n\)-Dimensional Relative Differential Geometry. arXiv:1712.10319v1 [math.DG] (2017)
  30. 30.
    Wang, Q.-M.: Isoparametric hypersurfaces in complex projective spaces. In: Differential Geometry and Differential Equations, Proceedings 1980 Beijing Sympososium, vol. 3, pp. 1509–1523 (1982)Google Scholar
  31. 31.
    Wang, Q.-M.: Isoparametric functions on Riemannian manifolds, I. Math. Ann. 277(277), 639–646 (1987)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Thorbergsson, G.: A Survey on Isoparametric Hypersurfaces and Their Generalizations, In Handbook of Differential Geometry, vol. I, pp. 963–995. North-Holland, Amsterdam (2000)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematics and Information SciencesHenan Normal UniversityXinxiangPeople’s Republic of China

Personalised recommendations

Cite article

Buy options