Frobenius manifolds occur naturally in the subject of symplectic topology, more specifically quantum cohomology.
A Riemannian manifold admits a compatible affine flat structure if and only if its curvature tensor vanishes everywhere.
Namely, the space of miniversal deformations of an isolated singularity has a Frobenius manifold structure.
This Frobenius manifold structure also relates to Kyoji Saito's primitive forms.
Manin, S.A. Merkulov: Semisimple Frobenius (super)manifolds and quantum cohomology of Pr, Topol.