  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.
