Abstract
In this paper, we study geometry of isometric minimal immersions of Riemannian surfaces in a symmetric space by moving frames and prove that the Gaussian curvature must be constant if the immersion is of parallel second fundamental form. In particular, when the surface is \(S^2\), we discuss the special case and obtain a necessary and sufficient condition such that its second fundamental form is parallel. We also consider isometric minimal two-spheres immersed in complex two-dimensional Kähler symmetric spaces with parallel second fundamental form, and prove that the immersion is totally geodesic with constant Kähler angle if it is neither holomorphic nor anti-holomorphic with Kähler angle \(\alpha \ne 0\) (resp. \(\alpha \ne \pi \)) everywhere on \(S^2\).
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Acknowledgements
This project was supported by the TianYuan Special Fund of the NSFC (Grant No. 11626220), the NSFC (Grant No. 11331002) and Startup Research Fund of Zhengzhou University (No. 1512315003). The authors would like to express their gratitude to the referee for his/her comments.
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Communicating Editor: Mj Mahan.
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Jiao, X., Li, M. Minimal surfaces in symmetric spaces with parallel second fundamental form. Proc Math Sci 127, 719–735 (2017). https://doi.org/10.1007/s12044-017-0352-5
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DOI: https://doi.org/10.1007/s12044-017-0352-5