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

Abstract

We show that a bi-flat 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 $d_{\nabla }{X}_{(\alpha +1)}=d_{L{\nabla }^{\ast }}{X}_{(\alpha )}$ 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.

19 pages