In this paper, we use the harmonic sequence to study the linearly full holomorphic two-spheres in complex Grassmann manifold $G(2,4)$. We show that if the Gaussian curvature 𝐾 (with respect to the induced metric) of a non-degenerate holomorphic two-sphere satisfies $K\leq 2$ (or $K\geq 2$), then 𝐾 must be equal to 2. Simultaneously, we show that one class of the holomorphic two-spheres with constant curvature 2 is totally geodesic. Concerning the degenerate holomorphic two-spheres, if its Gaussian curvature $K\leq 1$ (or $K\geq 1$), then $K=1$. Moreover, we prove that all holomorphic two-spheres with constant curvature 1 in $G(2,4)$ must be $U(4)$-equivalent.

For a harmonic map 𝑓 from a Riemann surface into a complex Grassmann manifold, Chern and Wolfson [4] constructed new harmonic maps $\partial f$ and $\overline{\partial} f$ through the fundamental collineations 𝜕 and $\overline{\partial}$ respectively. In this paper, we study the linearly full conformal minimal immersions from $S^2$ into complex Grassmannians $G(2,n)$, according to the relationships between the images of $\partial f$ and $\overline{\partial}f$. We obtain various pinching theorems and existence theorems about the Gaussian curvature, Kähler angle associated to the given minimal immersions, and characterize some immersions under special conditions. Some examples are given to show that the hypotheses in our theorems are reasonable.

In this paper, we study the non-degenerate holomorphic $S^2$ in the complex Grassmann manifold $G(k,n), 2k\leq n$, by the method of moving frame. For a non-degenerate holomorphic one, there exists globally defined positive functions $\lambda_1,\ldots,\lambda_k$ on $S^2$. We first show that the holomorphic $S^2$ in $G(k, 2k)$ is totally geodesic if these $\,\lambda_i$ are all equal. Conversely, for any totally geodesic immersion 𝑓 from $S^2$ into $G(k, n)$, we prove that $f(S^2)\subset G(k, 2k)$ up to $U(n)$-transformation, $\lambda_i=\frac{1}{\sqrt{k}}$, the Gaussian curvature $K=\frac{4}{k}$ and 𝑓 is given by $(z_0,z_1)\mapsto(z_0 I_k,z_1 I_k,0)$, in terms of homogeneous coordinate.

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 $S^2$.