• Minimal surfaces in symmetric spaces with parallel second fundamental form

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      Permanent link:
      https://www.ias.ac.in/article/fulltext/pmsc/127/04/0719-0735

    • Keywords

       

      Isometric minimal immersion; Gaussian curvature; Kähler angle; second fundamental form; symmetric space

    • 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 alsoconsider isometric minimal two-spheres immersed in complex two-dimensional Kählersymmetric spaces with parallel second fundamental form, and prove that the immersionis totally geodesic with constant Kähler angle if it is neither holomorphic nor antiholomorphicwith Kähler angle $\alpha\neq 0$ (resp. $\alpha\neq \pi$) everywhere on $S^2$.

    • Author Affiliations

       

      XIAOXIANG JIAO1 MINGYAN LI2

      1. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 101408, China
      2. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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