Reciprocal F-manifolds

We consider the action of a special class of reciprocal transformations on the principal hierarchy associated to a semisimple F-manifold with compatible flat structure (M, {ring operator} , ∇;, e) Under some additional assumptions, the hierarchy obtained applying these reciprocal transformations is also associated to an F-manifold with compatible flat structure that we call reciprocal F-manifold. We also consider the special case of bi-flat F-manifolds (M, {ring operator} , * , ∇; (1) , ∇; (2) , e, E) and we study reciprocal transformations preserving flatness of both the connections ∇; (1) and ∇; (2) and how they act on corresponding solutions of an augmented Darboux-Egorov system. © 2013 Elsevier B.V.