F-manifolds with flat structure and Dubrovin's duality

This work continues the study of F-manifolds (M,∘), first defined in [HeMa] (Int. Math. Res. Notices 6 (1999) 277-286, Preprint math.QA/9810132) and investigated in [He] (Frobenius Manifolds and Moduli Spaces for Singularities, Cambridge University Press, Cambridge, 2002). The notion of a compatible flat structure ∇ 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,∘,∇). In particular, we extend and generalize recent Dubrovin's duality ([Du2], On almost duality for Frobenius manifolds, Preprint math.DG/0307374).