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 $G(2, 4)$.