On the classification of homogeneous 2-spheres in complex Grassmannians

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 SU(2). First we observe that if an immersion x:S²→G(k+1,n+1) is homogeneous, then its image x(S²) is a 2-dimensional ρ(SU(2))-orbit in G(k+1,n+1), where ρ:SU(2)→U(n+1) is a unitary representation of 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).