Holomorphic Two-Spheres in the Complex Grassmann Manifold 𝐺(𝑘, 𝑛)

Proceedings Mathematical Sciences

Xiaowei Xu¹  Xu Zhong²  Xiaoxiang Jiao³ 

[1] $$;Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 00, People’s Republic of China$$;School of Mathematical Sciences, Graduate University of Chinese Academy of Sciences, Beijing 000, People’s Republic of China$$

关键词: Moving frame;    totally geodesic;    Gaussian curvature.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

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 $𝜆_1,ldots,𝜆_k$ 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 $f(S^2)subset G(k, 2k)$ up to $U(n)$-transformation, $𝜆_i=frac{1}{sqrt{k}}$, 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.

【 授权许可】

Unknown