Skip to main content
SpringerLink
Log in

Harmonic polynomial morphisms between Euclidean spaces

  • Published:
Science in China Series A: Mathematics Aims and scope Submit manuscript

Abstract

Two non-existence theorems on harmonic polynomial morphisms between Euclidean spaces have been shown.

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. Eells, J., Lemaire, L., A report on harmonic maps,Bull. London Math. Soc., 1978, 10: 1.

    Article  MathSciNet  MATH  Google Scholar 

  2. Baird, P., Harmonic maps with symmetry, harmonic morphisms and deformations of metrics, inRes. Notes in Math., Vol. 87, London: Pitman, 1983.

    Google Scholar 

  3. Fuglede, B., Harmonic morphisms between Riemannian manifolds,Ann. Inst. Fourier (Grenoble), 1978, 28: 107.

    Article  MathSciNet  MATH  Google Scholar 

  4. Ishihara, T., A mapping of Riemannian manifolds which preserves harmonic functions,J. Math. Kyoto Univ., 1979, 12: 215.

    MathSciNet  Google Scholar 

  5. Bernard, A., Campbell, E. A., Davie, A. M., Brownian motion and generalized analytic and inner functions,Ann. Inst. Fourier (Grenoble), 1979, 29: 207.

    Article  MathSciNet  MATH  Google Scholar 

  6. Baird, P., Harmonic morphisms and circle action on 3- and 4-manifolds,Ann. Inst. Fourier (Grenoble), 1990, 40: 177.

    Article  MathSciNet  MATH  Google Scholar 

  7. Baird, P., Ou, Y. L., Harmonic maps and morphisms, from multilinear norm-preserving mappings, Inter. J. Math., 1997, 187.

  8. Eells, J., Yiu, P., Polynomial harmonic morphisms between Euclidean spheres,Proc. A. M. S., 1995, 123: 2921.

    Article  MathSciNet  MATH  Google Scholar 

  9. Lam, K. Y., Some new results in compositions of quadratic forms,Invent. Math., 1985, 79: 467.

    Article  MathSciNet  MATH  Google Scholar 

  10. Yiu, P., Quadratic forms between spheres and the non-existence of sums of squares formulae,Math. Proc. Comb. Phil. Soc., 1986, 100: 493.

    Article  MathSciNet  MATH  Google Scholar 

  11. Ginsburg, M., Some immersions of projective space in Euclidean space,Topology, 1963, 2: 69.

    Article  MathSciNet  MATH  Google Scholar 

  12. Münzner, H. F., Isoparametric hyperflachen in spharen,I Math. Ann., 1980, 251: 57.

    Article  MATH  Google Scholar 

  13. Münzner, H. F., Isoparametric hyperflachen in spharen,II Math. Ann., 1981, 256: 215.

    Article  MATH  Google Scholar 

  14. Ou, Y. L., Quadratic harmonic morphisms and O-systemes,Ann. Inst. Fourier (Grenoble), 1997, 47: 687.

    Article  MathSciNet  MATH  Google Scholar 

  15. Nomizu, K., Eliee Cartan’s work on isoparametric hypersurfaces in spheres,Proc. Symp. Pure Math., 1975, 27: 191.

    MathSciNet  Google Scholar 

  16. Adams, J. F., Vector fields on spheres,Ann. Math., 1962, 75: 603.

    Article  MATH  Google Scholar 

  17. Eckmann, B., Gruppentheorelischer beweis des satzes von Hurwitz-Radon uber die komposition quadratischer formen,Comm. Math. Helv., 1943, 15: 358.

    Article  MATH  Google Scholar 

  18. Adams, J. F., Lax, P. D., Philips, R. S., On matrices whose linear combinations are nonsingular,Proc. A. M. S., 1965, 16: 318.

    Google Scholar 

  19. Adams, J. F., Lax, P. D., Phillips, R. S., On matrices whose linear combinations are nonsingular,Proc. A. M. S., 1966, 17: 945.

    MathSciNet  Google Scholar 

  20. Ferus, D., Karcher, H., Munzner, H. F., Clifford algebren und neue isoparametrische hyperflachen,Math. Z., 1981, 177: 479.

    Article  MathSciNet  MATH  Google Scholar 

  21. Cartan, E., Sur des families remarquables d’hypersurfaces isoparametriques dans les espaces spheriques,Math. Z., 1939, 45: 335.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Tang, Z. Z., Isoparametric hypersurfaces with four distinct principal curvatures,Chinese Sci. Bull., 1991, 36: 1237.

    MathSciNet  MATH  Google Scholar 

  24. Dorfmesiter, J., Neher, E., Isoparametric hypersurfaces, case g = 6, m = 1,Comm. in Alg., 1985, 13: 2299.

    Article  Google Scholar 

  25. Takagi, R., Takahashi, T., On the principal curvatures of homogeneous hypersurfaces in a sphere, inDifferential Geometry, in honor of K. Yano (ed. Kabayashi, S.), Tokyo: Kinokuniya, 1972, 469.

    Google Scholar 

  26. Peng, C. K., Hou, Z. X., A remark on the isoparametric polynomial of degree 6, in differential geometry and topology,Lectures Notes in Math. (ed. Jiang, B.) Vol. 1369, Berlin: Springer-Verlag, 1986-87, 222.

    Google Scholar 

  27. Miyaoka, R., The linear isotropy group of G2/SO(4), the Hopf fibering and isoparametric hypersurfaces,Osaka J. Math., 1993, 30: 179.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Tsinghua University, 100084, Beijing, China

    Tang Zizhou

Authors
  1. Tang Zizhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Project supported partially by the National Natural Science Foundation of China (Grant No. 19531050) and the State Education Commission Foundation of China.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zizhou, T. Harmonic polynomial morphisms between Euclidean spaces. Sci. China Ser. A-Math. 42, 570–576 (1999). https://doi.org/10.1007/BF02880074

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02880074

Keywords

Search

Navigation