A geometry of cubic discriminants in 8 dimensions

TL;DR

This paper explores 8-dimensional manifolds with special geometric structures linked to an irreducible SO(4) representation, introducing a cubic discriminant tensor and classifying certain symmetric spaces with unique curvature properties.

Contribution

It introduces a cubic discriminant tensor field characterizing these manifolds and classifies the non-flat, integrable examples as specific quaternion-Kähler symmetric spaces.

Findings

Identification of a cubic discriminant tensor field for these structures

Classification of non-flat, integrable examples as specific symmetric spaces

New curvature-based characterization of the metrics on these spaces

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-K\"ahler symmetric spaces $G_2\big/SO(4)$ and $G_{2(2)}\big/SO(4)$. We also present a new curvature-based characterization for the Riemannian metrics on these spaces.