Extrinsic homogeneous almost Hermitian 6-dimensional submanifolds in the octonions (Q2480070)

The octonions (or Cayley algebra) \(\mathfrak C\) over \(\mathbb R\) is a direct sum of quaternions \(H\oplus H=\mathfrak C\) with the following multiplication \[ (a+b\varepsilon)(c+d\varepsilon)= ac-\bar d b+(da+b\bar c)\varepsilon \] where \(a,b,c,d\in H\) and \(\varepsilon=(0,1)\in H\oplus H\). R. L. Bryant has shown that any oriented 6-dimensional submanifold of the octonions admits a Spin (7)-invariant almost complex structure. Extrinsic homogeneous submanifolds of \(\mathfrak C\) are defined as submanifolds obtained as the orbit of a Lie subgroup of \(\mathbb R^{8}\rtimes \text{Spin}(7)\). In this paper, the authors give a classification of 6-dimensional extrinsic homogeneous almost Hermitian submanifolds of \(\mathfrak C\) by making use of the classification of the homogeneous isoparametric hypersurfaces of a unit sphere. The main result is theorem 5.1. They also introduce a list of 6-dimensional Riemannian homogeneous submanifolds of \(\mathfrak C\) which are not homogeneous with respect to the induced almost complex structure.