Holomorphic two-spheres in the complex Grassmann manifold G(k, n)

Abstract

In this paper, we study the non-degenerate holomorphic S 2 in the complex Grassmann manifold G(k, n), 2k ≤ n, by the method of moving frame. For a non-degenerate holomorphic one, there exists globally defined positive functions λ 1, ..., λ k on S 2. We first show that the holomorphic S 2 in G(k, 2k) is totally geodesic if these λ i are all equal. Conversely, for any totally geodesic immersion f from S 2 into G(k, n), we prove that f(S 2) ⊂ 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)↦(z 0 I k , z 1 I k , 0), in terms of homogeneous coordinate.