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.

中文翻译：

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$F$-流形、多平面结构和Painlevé超验

在本文中，我们研究了配备多个平面连接（和多个 $F$ 产品）的 $F$-歧管，这些连接件需要在适当的意义上兼容。在半简单的情况下，我们表明存在这种多个平面连接的必要条件可以用矢量场分布的可积性来表达，这些矢量场分布与所涉及的多个产品的最终身份相关。使用这个事实，我们表明通常不会有超过三个平面连接的多平面结构。当相关分布可积时，我们在维度 $2$ 和 $3$ 上构建双平面 $F$-流形，在维度 $3$ 和 $4$ 上构建三平面 $F$-流形。特别地，我们根据可以简化为 P$_{VI}$ 方程的完整族的六个一阶 ODE 系统获得了三维双平面 $F$ 的参数化，并且我们构造了非平凡的例子由超几何函数控制的四维三平面 $F$ 流形。在论文的第二部分，我们将分析扩展到包括非半简单的多平面 $F$-流形。我们表明，在维度三中，根据 Jordan 范式，规则的非半简单双平面 $F$-流形通过完整的 P$_{IV}$ 和 P$_{V}$ 方程的解局部参数化自同态 $L=E\circ$。将此结果与 $3$-维双平面 $F$-流形的局部参数化相结合，我们得到 P$_{IV}$ 的汇合，P$_{V}$ 和 P$_{VI}$ 对应于 $L$ 的特征值的碰撞，保留了规律性。此外，我们表明，与半简单情况相反，可以构建具有任意数量兼容平面连接的常规非半简单多平面 $F$-流形。