F-manifolds with flat structure and Dubrovin's duality

This work continues the study of $F$–manifolds $(M,\circ )$, first defined by Hertling and Manin and investigated in [He]. The notion of a compatible flat structure $\mathrm{\nabla }$ is introduced, and it is shown that many constructions known for Frobenius manifolds do not in fact require invariant metrics and can be developed for all such triples $(M,\circ ,\mathrm{\nabla })\mathrm{.}$ In particular, we extend and generalize recent Dubrovin’s duality.