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arXiv · 2508.12014

A geometry of cubic discriminants in 8 dimensions

Elitza Hristova·Ivan Minchev
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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.

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Elitza Hristova, Ivan Minchev. 2025-08-16. A geometry of cubic discriminants in 8 dimensions. https://arxiv.org/abs/2508.12014

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