Skip to main content
Log in

Rigidity of holomorphic curves in complex Grassmann manifolds

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract.

In this paper it is proved that the holomorphic curves in complex Grassmann manifolds are uniquely determined by its first and second fundamental forms, up to rigid motion.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Calabi, E.: Isometric embedding of complex manifolds. Ann. Math. 58, 1–23 (1953)

    MathSciNet  MATH  Google Scholar 

  2. Chi, Q., Zheng, Y.: Rigidity of pseudo-holomorphic curves of constant curvature in Grassmann manifolds. Trans. Am. Math. Soc. 313, 393–406 (1989)

    MathSciNet  MATH  Google Scholar 

  3. Griffiths, P.: On Cartan’s method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry. Duke Math. J. 41, 775–814 (1974)

    MATH  Google Scholar 

  4. Helgason, S.: Differential geometry. Lie groups, and symmetric spaces, New York, Academic Press, 1978

  5. Jiao, X.X.: Finite harmonic maps of surfaces into U(n) (Chinese). Adv. Math. (China) 27, 533–535 (1998)

    MathSciNet  MATH  Google Scholar 

  6. Jiao, X.X., Peng, J.G.: Classification of holomorphic two-spheres with constant curvature in complex Grassmann manifold G 2,5. Differ. Geom. Appl., to appear

  7. Lawson, H.B.: The Riemannian geometry of holomorphic curves, Proc. Conf. On holomorphic mapping and minimal surfaces. Bol. Soc. Brazil. Mat. 2, 45–62 (1971)

    MATH  Google Scholar 

  8. Uhlenbeck, K.: Harmonic maps into Lie groups (classical solutions of the chiral model). J. Diff. Geom. 30, 1–50 (1989)

    MathSciNet  MATH  Google Scholar 

  9. Zheng, Y.B.: Quantization of curvature of harmonic two-spheres in Grassmann manifolds. Trans. Amer. Math. Soc. 316, 193–214 (1989)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Xiaoxiang Jiao.

Additional information

Supported by the National Natural Science Foundation of China and the President Foundation of Graduate School of the Chinese Academy of Sciences.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jiao, X., Peng, J. Rigidity of holomorphic curves in complex Grassmann manifolds. Math. Ann. 327, 481–486 (2003). https://doi.org/10.1007/s00208-003-0462-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-003-0462-5

Keywords

Navigation