Classification of homogeneous holomorphic two-spheres in complex Grassmann manifolds

In this paper we completely classify the linearly full homogeneous holomorphic two-spheres in the complex Grassmann manifolds $G(2,N)$ and $G(3,N)$. We also obtain the Gauss equation for the holomorphic immersions from a Riemann surface into $G(k,N)$. By using which, we give explicit expressions of the Gaussian curvature and the square of the length of the second fundamental form of these homogeneous holomorphic two-spheres in $G(2,N)$ and $G(3,N)$.