$F$-manifolds, multi-flat structures and Painlev\'e transcendents

In this paper we study $F$-manifolds equipped with multiple flat connections (and multiple $F$-products), that are required to be compatible in a suitable sense. In the semisimple case we show that a necessary condition for the existence of such multiple flat connections can be expressed in terms of the integrability of a distribution of vector fields that are related to the eventual identities for the multiple products involved. Using this fact we show that in general there can not be multi-flat structures with more than three flat connections. When the relevant distributions are integrable we construct bi-flat $F$-manifolds in dimension $2$ and $3$, and tri-flat $F$-manifolds in dimensions $3$ and $4$. In particular we obtain a parametrization of three-dimensional bi-flat $F$ in terms of a system of six first order ODEs that can be reduced to the full family of P$_{VI}$ equation and we construct non-trivial examples of four dimensional tri-flat $F$ manifolds that are controlled by hypergeometric functions. In the second part of the paper we extend our analysis to include non-semisimple multi-flat $F$-manifolds. We show that in dimension three, regular non-semisimple bi-flat $F$-manifolds are locally parameterized by solutions of the full P$_{IV}$ and P$_{V}$ equations, according to the Jordan normal form of the endomorphism $L=E\circ$. Combining this result with the local parametrization of $3$-dimensional bi-flat $F$-manifolds we have that confluences of P$_{IV}$, P$_{V}$ and P$_{VI}$ correspond to collisions of eigenvalues of $L$ preserving the regularity. Furthermore, we show that contrary to the semisimple situation, it is possible to construct regular non-semisimple multi-flat $F$-manifolds, with any number of compatible flat connections.