Skip to main content
SpringerLink
Log in

On holomorphic curves in a complex Grassmann manifold G(2, n)

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

Let s : S 2 → G(2, n) be a linearly full totally unramified non-degenerate holomorphic curve in a complex Grassmann manifold G(2, n), and let K(s) be its Gaussian curvature. It is proved that \({K(s) = \frac{4}{n-2}}\) if K(s) satisfies \({K(s) \geq \frac{4}{n-2}}\) or \({K(s) \leq \frac{4}{n-2} }\) everywhere on S 2. In particular, \({K(s) = \frac{4}{n-2}}\) if K(s) is constant.

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. Bolton J. et al.: On conformal minimal immersions of S2 into CP n, Math. Ann. 279, 599–620 (1988)

    MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Chern S.S., Wolfson J.G.: Harmonic maps of the two-sphere in a complex Grassmann manifold. Ann. Math. 125, 301–335 (1987)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Jiao X.X.: Pseudo-holomorphic curves of constant curvature in complex Grassmann. Israel J. Math. 163, 45–60 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Jiao X.X.: On pseudo-holomorphic curves from two-spheres into a complex Grassmannian G(2, 5). Acta Math. Sinica 26, 759–762 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. Jiao X.X., Peng J.G.: Pseudo-holomorphic curves in complex Grassmann manifolds. Trans. Amer. Math. Soc. 355, 3715–3726 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Rigoli M.: A rigidity result for holomorphic emmersions of surfaces in CP n. Proc. Amer. Math. Soc. 93, 317–320 (1985)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Graduate University, Chinese Academy of Sciences, Beijing, 100049, China

    Xiaoxiang Jiao

  2. College of Sciences, Northeast Dianli University, Jilin City, 132012, China

    Yan Yu

Authors
  1. Xiaoxiang Jiao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yan Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaoxiang Jiao.

Additional information

This work is supported by the National Natural Science Foundation of China (Grant No. 11071248) and Knowledge Innovation Funds of CAS (Grant No. KJCX3-SYW-S03).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jiao, X., Yu, Y. On holomorphic curves in a complex Grassmann manifold G(2, n). Arch. Math. 96, 291–300 (2011). https://doi.org/10.1007/s00013-011-0231-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-011-0231-8

Mathematics Subject Classification (2010)

Keywords

Search

Navigation