On fibre bundle structures of Stiefel manifolds related to the octonions

In this paper, we investigate the real Stiefel manifolds Vk(Rn)=SO(n)/SO(n−k) for n=7 or n=8. If we identify R7 and R8 with ImC and C, respectively, then we have V2(R7)=G2/SU(2), V2(R8)=Spin(7)/SU(3), and V3(R8)=Spin(7)/SU(2). These real Stiefel manifolds can be represented as an orbit of the action of the Lie group G2 or Spin(7). In this paper, we give the orbit decompositions of Stiefel manifolds related to the octonions C under the action of Lie group G2 and Spin(7).