Isometric minimal immersion; Gaussian curvature; Kähler angle; second fundamental form; symmetric space
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 $S^2$.