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

We show that a bi-flat F-structure $(\nabla,\circ,e,\nabla^*,*,E)$ on a manifold $M$ defines a differential bicomplex $(d_{\nabla},d_{E\circ\nabla^*})$ on forms with value on the tangent sheaf of the manifold. Moreover, the sequence of vector fields defined recursively by $d_{\nabla}X_{(\alpha+1)}=d_{L\nabla^*}X_{(\alpha)}$ coincide with the coefficients of the formal expansion of the flat local sections of a family of flat connections $\nabla^{GM}$ associated with the bi-flat structure. In the case of Dubrovin-Frobenius manifold the connection $\nabla^{GM}$ (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.