Bi-flat F-structures as differential bicomplexes and Gauss-Manin connections

We show that a bi-flat F-structure (, , e, ^*, *, E) on a manifold M defines a differential bicomplex (d_, d₄^*) on forms with value on the tangent sheaf of the manifold. Moreover, the sequence of vector fields defined recursively by d_X (+₁) =d₋^*X () coincide with the coefficients of the formal expansion of the flat local sections of a family of flat connections ^GM associated with the bi-flat structure. In the case of Dubrovin-Frobenius manifold the connection ^GM (for suitable choice of an auxiliary parameter) can be identified with the Levi-Civita connection of the flat pencil of metrics defined by the invariant metric and the intesection form.