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(kC1, nC1) by theory of unitary representations of the 3-dimensional special unitary group SU(2). First we observe that if an immersion x W S＾2 ! G(k C 1, n C 1) is homogeneous, then its image x(S＾2) is a 2-dimensional _(SU(2))-orbit in G(k C1, nC1), where _W SU(2)!U(nC1) is a unitary representation of SU(2). Then we give a classification theorem of homogeneous 2-spheres in G(k C1, n C1). 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).