Osaka Journal of Mathematics

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

Jie Fei, Xiaoxiang Jiao, Liang Xiao, and Xiaowei Xu

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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

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

First available in Project Euclid: 27 March 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42] 53C55: Hermitian and Kählerian manifolds [See also 32Cxx]


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.

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