A geometry of cubic discriminants in 8 dimensions

Abstract

This paper examines 8-dimensional Riemannian manifolds whose structure group reduces to ${SO(4)}_{ir}\subset GL(8,\mathbb{R})$, the image of an irreducible representation of $SO(4)$ on ${\mathbb{R}}^8$. 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 $G_2/SO(4)$ and $G_{2(2)}/SO(4)$. We also present a new curvature-based characterization for the Riemannian metrics on these spaces.