From Darboux-Egorov system to bi-flat $F$-manifolds

Motivated by the theory of integrable PDEs of hydrodynamic type and by the generalization of Dubrovin's duality in the framework of $F$-manifolds due to Manin [22], we consider a special class of $F$-manifolds, called bi-flat $F$-manifolds. A bi-flat $F$-manifold is given by the following data $(M, \nabla_1,\nabla_2,\circ,*,e,E)$, where $(M, \circ)$ is an $F$-manifold, $e$ is the identity of the product $\circ$, $\nabla_1$ is a flat connection compatible with $\circ$ and satisfying $\nabla_1 e=0$, while $E$ is an eventual identity giving rise to the dual product *, and $\nabla_2$ is a flat connection compatible with * and satisfying $\nabla_2 E=0$. Moreover, the two connections $\nabla_1$ and $\nabla_2$ are required to be hydrodynamically almost equivalent in the sense specified in [2]. First we show that, similarly to the way in which Frobenius manifolds are constructed starting from Darboux-Egorov systems, also bi-flat $F$-manifolds can be built from solutions of suitably augmented Darboux-Egorov systems, essentially dropping the requirement that the rotation coefficients are symmetric. Although any Frobenius manifold possesses automatically the structure of a bi-flat $F$-manifold, we show that the latter is a strictly larger class. In particular we study in some detail bi-flat $F$-manifolds in dimensions n=2, 3. For instance, we show that in dimension 3 bi-flat $F$-manifolds are parametrized by solutions of a two parameters Painlev\'e VI equation, admitting among its solutions hypergeometric functions. Finally we comment on some open problems of wide scope related to bi-flat $F$-manifolds.