Lagrangian submanifolds of $S^6$ and the associative Grassmann manifold

Abstract

We focus on Lagrangian submanifolds of a six-dimensional sphere in the space Im $\mathbf {O}$ of imaginary octonions and study the relationship of such submanifolds with the geometry of the associative Grassmann manifold $\widetilde{{\rm Gr}}_{ass}$(Im $\mathbf {O}$) which is the Grassmann manifold of associative subspaces in Im $\mathbf {O}$. Considering the Gauss maps into $\widetilde{{\rm Gr}}_{ass}$(Im $\mathbf {O}$) associated to Lagrangian submanifolds, we show that those maps are harmonic. Moreover, the Gauss maps associated to homogeneous Lagrangian submanifolds are investigated.