Lagrangian submanifolds of S⁶ and the associative Grassmann manifold

Abstract

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