Abstract
In this paper, we give a classification theorem of minimal two-spheres in G(2, 4; (C)) with parallel second fundamental form. Moreover, we also consider some special holomorphic two-spheres in G(2, n; (C)) and give the corresponding conditions of the parallel second fundamental form.
Similar content being viewed by others
References
S. S. Chern and J. G. Wolfson, “Minimal surfaces by moving frames,” Am. J. Math. 105, 59–83 (1983).
J. Fei, X. X. Jiao, L. Xiao, and X. W. Xu, “On the classification of homogeneous 2-spheres in complex Grassmannians,” Osaka J. Math. 50, 135–152 (2013).
L. He and X. X. Jiao, “Classification of all conformal minimal immersions of S2 in HP n with parallel second fundamental form,” Annali diMatematica 194, 1301–1317 (2015).
X. X. Jiao, “On minimal two-spheres immersed in complex Grassmann manifolds with parallel second fundamental form,” Monatsh. Math. 168, 381–401 (2012).
X. X. Jiao and J. G. Peng, “On some conformal minimal two-spheres in a complex projective space,” Quart. J. Math. 61 (1), 87–101 (2010).
X. X. Jiao and J. G. Peng, “Minimal two-spheres in G(2, 4),” Front. Math. China 5 (2), 297–310 (2010).
X. X. Jiao and J. Wang, “Minimal surfaces in a complex hyperquadric Q 2,” ManuscriptaMath. 140, 597–611 (2013).
X. X. Jiao and Y. Yan, “On holomorphic curves in a complex Grassmann manifold G(2, n),” Arch. Math. 96, 291–300 (2011).
Z. Q. Li and Z. H. Yu, “Constant curved minimal two-spheres in G(2, 4),” Manuscr. Math. 100, 305–316 (1999).
H. Nakagawa and R. Takagi, “On locally symmetric Kaehler submanifolds in a complex projective space,” J. Math. Soc. Jpn. 28 (4), 638–667 (1976).
Y. B. Shen, “The Riemannian geometry of superminimal surfaces in complex space forms,” Acta Math. Sini. 12 (3), 298–313 (1996).
K. Uhlenbeck, “Harmonic maps into Lie groups (classical solutions of the chiral model),” J. Differ. Geom. 30, 1–50 (1989).
Author information
Authors and Affiliations
Corresponding author
Additional information
The article was submitted by the authors for the English version of the journal.
Rights and permissions
About this article
Cite this article
Zhang, W., Jiao, X. & Li, M. Minimal two-spheres in G(2, 4; (C)) with parallel second fundamental form. Math Notes 101, 899–912 (2017). https://doi.org/10.1134/S0001434617050157
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434617050157