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This is a Quantitative Study study.

September 24, 2025Open Access

A geometry of cubic discriminants in 8 dimensions

EHElitza Hristova IMIvan MinchevLeibniz Institute for Astrophysics Potsdam

Key Points

Cubic discriminants relate to structures of 8-dimensional Riemannian manifolds, which can be described using specific tensor fields.
The analysis reveals that non-flat, integrable structures behave as quaternion-Kähler symmetric spaces under certain conditions.
An almost quaternion-Hermitian structure is key in understanding these Riemannian manifolds and how they relate to classical concepts.
The paper proposes a new curvature-based characterization that could enhance the understanding of metrics on these manifolds.

Abstract

This paper examines 8-dimensional Riemannian manifolds whose structure group reduces to SO (4) ₈ₑ 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.

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Cite This Study

Hristova et al. (Sat,) studied this question.

synapsesocial.com/papers/68d6e1978b2b6861e4c40361 https://doi.org/https://doi.org/10.48550/arxiv.2508.12014

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