A perspective on totally geodesic submanifolds of the symmetric space G₂/SO(4)

We provide an independent proof of the classification of the maximal totally geodesic submanifolds of the symmetric spaces G₂ and G₂/SO(4), jointly with very natural descriptions of all of these submanifolds. The description of the totally geodesic submanifolds of G₂ is in terms of (1) principal subalgebras of g₂; (2) stabilizers of nonzero points of R⁷; (3) stabilizers of associative subalgebras; (4) the set of order two elements in G₂ (and its translations). The space G₂/SO(4) is identified with the set of associative subalgebras of R⁷ and its maximal totally geodesic submanifolds can be described as the associative subalgebras adapted to a fixed principal subalgebra, the associative subalgebras orthogonal to a fixed nonzero vector, the associative subalgebras containing a fixed nonzero vector, and the associative subalgebras intersecting both a fixed associative subalgebra and its orthogonal. A second description is included in terms of Grassmannians, the advantage of which is that the associated Lie triple systems are easily described in matrix form.

• https://arxiv.org/abs/2504.07586