On Frobenius structures in symmetric cones

We prove that in any strictly convex symmetric cone $\mathrm{\Omega }$ there exists a non empty locus where the WDVV equation is satisfied (i.e. there exists a hyperplane being a Frobenius manifold). This result holds over any real division algebra (with a restriction to the rank 3 case if we consider the field $\mathbb{O}$) but also on their linear combinations. This theorem holds as well in the case of pseudo-Riemannian geometry, in particular for a Lorentz symmetric cone of Anti-de-Sitter type. Our statement can be considered as a generalisation of a result by Ferapontov–Kruglikov–Novikov and Mokhov. Our construction is achieved by merging two different approaches: an algebraic/geometric one and the analytic approach given by Calabi in his investigations on the Monge–Ampère equation for the case of affine hyperspheres.