Biflat F-structures as differential bicomplexes and Gauss–Manin connections

Abstract

We show that a biflat F-structure $(\nabla ,\circ ,e,{\nabla }^{\ast },\ast ,E)$ on a manifold $M$ defines a differential bicomplex $(d_{\nabla },d_{E\circ {\nabla }^{\ast }})$ on forms with value on the tangent sheaf of the manifold. Moreover, the sequence of vector fields defined recursively by $d_{\nabla }{X}_{(\alpha +1)}=d_{E\circ {\nabla }^{\ast }}{X}_{\left(\alpha \right)}$ coincides with the coefficients of the formal expansion of the flat local sections of a family of flat connections ${\nabla }^{GM}$ associated with the biflat 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 intersection form.