Metricity of strongly non-degenerate 4-forms

Determine whether every metric 4-form on an 8-dimensional real vector space is strongly non-degenerate.

Background

The paper distinguishes metric 4-forms, defined through the non-degeneracy of the quadratic form associated with q_α, from strongly non-degenerate 4-forms, defined by definiteness of H_α. The constructed family β_t contains forms that are metric but, for certain parameter values, not strongly non-degenerate.

This demonstrates that strong non-degeneracy does not imply metricity. The converse implication—whether metricity implies strong non-degeneracy—remains unresolved.

References

Outside of this interval, and for t\neq0, \beta_t is still a metric 4-form, per the result in eq:gbeta. Thus strong non-degeneracy does not imply that a 4-form is metric\footnote{The converse may be true, but we could not prove it.}.

— A note on 4-forms in 8-dimensions  (2608.20200 - Close, 20 Aug 2026) in Section 3, final paragraph before Acknowledgements