A geometry of cubic discriminants in 8 dimensions

This paper examines 8-dimensional Riemannian manifolds whose structure group reduces to SO(4)ᵢᵣ⊂ GL(8, R), the image of an irreducible representation of $SO(4)$ on R⁸. We demonstrate that such a reduction can be described by an almost quaternion-Hermitian structure and a special rank-4 tensor field, which we call a cubic discriminant. This tensor field is pointwise linearly equivalent to the formula for the discriminant of a cubic polynomial. We show that the only non-flat, integrable examples of these structures are the quaternion-Kähler symmetric spaces G₂/SO(4) and G₂₍₂₎/SO(4). We also present a new curvature-based characterization for the Riemannian metrics on these spaces.