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Acta Mathematica Scientia

, Volume 39, Issue 1, pp 312–328 | Cite as

Classfication of Homogeneous Two-Spheres in G(2, 5;C)

  • Wenjuan Zhang (张文娟)
  • Jie Fei (费杰)Email author
  • Xiaoxiang Jiao (焦晓祥)
Article

Abstract

In this article, we determine all homogeneous two-spheres in the complex Grassmann manifold G(2, 5;C) by theory of unitary representations of the 3-dimensional special unitary group SU(2).

Key words

homogeneous immersion Gauss curvature Kähler angle rigidity 

2010 MR Subject Classification

Primary 53C42 53C55 

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References

  1. [1]
    Bolton J, Jensen G R, Rigoli M, Woodward L M. On conformal minimal immersions of S2 into CPn. Math Ann, 1988, 279: 599–620MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Bryant R L. Minimal surfaces of constant curvature in Sn. Amer Math, 1985, 290: 259–271MathSciNetzbMATHGoogle Scholar
  3. [3]
    Bando S, Ohnita Y. Minimal 2-spheres with constant curvature in Pn(C). J Math, 1987, 39: 477–487zbMATHGoogle Scholar
  4. [4]
    Calabi E. Minimal immersions of surfaces in Euclidean spheres. J Diff Geom, 1967, 1: 111–125MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Chern S S, Wolfson J G. Harmonic maps of the two-sphere in a complex Grassmann manifold. Ann Math, 1987, 125: 301–335MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Chern S S, Wolfson J G. Minimal surfaces by moving frames. Amer J Math, 1983, 105: 59–83MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Chi Q S, Zheng Y. Rigidity of pseudo-holomorphic curves of constant curvature in Grassmann manifolds. Amer Math, 1989, 313: 393–406MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Fei J, Jiao X X, Xiao L, Xu XW. On the classification of homogeneous 2-spheres in complex Grassmannians. Osaka J Math, 2013, 50: 135–152MathSciNetzbMATHGoogle Scholar
  9. [9]
    Fei J, Jiao X X, Xu X W. Construction of homogeneous minimal 2-spheres in complex Grassmannians. Acta Math Sci, 2011, 31B(4): 1889–1898MathSciNetzbMATHGoogle Scholar
  10. [10]
    He L, Jiao X X, Zhou X C. Rigidity of holomorphic curves of constant curvature in G(2, 5). Diff Geom Appl, 2015, 43: 21–44MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Jiao X X, Peng J G. Classification of holomorphic spheres of constant curvature in complex Grassmann manifold G2,5. Diff Geom Appl, 2004, 20: 267–277CrossRefzbMATHGoogle Scholar
  12. [12]
    Kenmotsu K. On minimal immersions of R 2 into S N. J Math, 1976, 28: 182–191MathSciNetzbMATHGoogle Scholar
  13. [13]
    Kobayashi S, Nomizu K. Foundations of Differential Geometry. II. New York: Interscience Publishers John Wiley Sons, Inc, 1969zbMATHGoogle Scholar
  14. [14]
    Li H, Wang C, Wu F. The classification of homogeneous 2-spheres in CP n. Asian J Math, 2001, 5: 93–108MathSciNetCrossRefGoogle Scholar
  15. [15]
    Li M Y, Jiao X X, He L. Classification of conformal minimal immersions of constant curvature from S 2 to Q 3. J Math Soc Japan, 2016, 68: 863–883MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Li Z Q, Yu Z H. Constant curved minimal 2-spheres in G(2, 4). Manuscripta Math, 1999, 100: 305–316MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Ogiue K. Differential geometry of Käehler submanifolds. Adv Math, 1974, 13: 73–114CrossRefzbMATHGoogle Scholar

Copyright information

© Wuhan Institutes of Physics and Mathematics, Chinese Academy of Sciences 2019

Authors and Affiliations

  • Wenjuan Zhang (张文娟)
    • 1
  • Jie Fei (费杰)
    • 2
    Email author
  • Xiaoxiang Jiao (焦晓祥)
    • 3
  1. 1.School of ScienceEast China University of TechnologyNanchangChina
  2. 2.Department of Mathematical SciencesXi’an Jiaotong-Liverpool UniversitySuzhouChina
  3. 3.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina

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