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https://www.ias.ac.in/article/fulltext/pmsc/122/03/0399-0409

Moving frame; totally geodesic; Gaussian curvature.

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.

Xiaoxiang Jiao¹ Xu Zhong¹ Xiaowei Xu²

1. School of Mathematical Sciences, Graduate University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China
2. Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China

Published

August 2012

Manuscript received

3 January 2011

Manuscript revised

27 May 2012