## Osaka Journal of Mathematics

### On the classification of homogeneous $2$-spheres in complex Grassmannians

#### Abstract

In this paper we discuss a classification problem of homogeneous 2-spheres in the complex Grassmann manifold $G(k + 1, n + 1)$ by theory of unitary representations of the 3-dimensional special unitary group $\mathit{SU}(2)$. First we observe that if an immersion $x\colon S^{2} \to G(k + 1, n + 1)$ is homogeneous, then its image $x(S^{2})$ is a 2-dimensional $\rho(\mathit{SU}(2))$-orbit in $G(k + 1, n + 1)$, where $\rho\colon \mathit{SU}(2) \to U(n + 1)$ is a unitary representation of $\mathit{SU}(2)$. Then we give a classification theorem of homogeneous 2-spheres in $G(k + 1, n + 1)$. As an application we describe explicitly all homogeneous 2-spheres in $G(2, 4)$. Also we mention about an example of non-homogeneous holomorphic 2-sphere with constant curvature in $G(2, 4)$.

#### Article information

Source
Osaka J. Math., Volume 50, Number 1 (2013), 135-152.

Dates
First available in Project Euclid: 27 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1364390422

Mathematical Reviews number (MathSciNet)
MR3080633

Zentralblatt MATH identifier
1263.53052

#### Citation

Fei, Jie; Jiao, Xiaoxiang; Xiao, Liang; Xu, Xiaowei. On the classification of homogeneous $2$-spheres in complex Grassmannians. Osaka J. Math. 50 (2013), no. 1, 135--152. https://projecteuclid.org/euclid.ojm/1364390422

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