Darboux-Egorov system, bi-flat $F$-manifolds and Painlev\'e VI

TL;DR

This paper generalizes a method for constructing semisimple bi-flat F-manifolds from solutions of the Darboux-Egorov system, revealing their connection to Painlevé VI equations in three dimensions.

Contribution

It extends previous constructions to cases with different degrees of homogeneity and links three-dimensional bi-flat F-manifolds to Painlevé VI solutions.

Findings

Construction of bi-flat F-manifolds from homogeneous Darboux-Egorov solutions

Homogeneity degrees of Lamé and rotation coefficients generalized

Three-dimensional bi-flat F-manifolds parametrized by Painlevé VI solutions

Abstract

This is a generalization of the procedure presented in [3] to construct semisimple bi-flat $F$-manifolds $(M,\nabla^{(1)},\nabla^{(2)},\circ,*,e,E)$ starting from homogeneous solutions of degree -1 of Darboux-Egorov-system. The Lam\'e coefficients $H_i$ involved in the construction are still homogeneous functions of a certain degree $d_i$ but we consider the general case $d_i\ne d_j$. As a consequence the rotation coefficients $\beta_{ij}$ are homogeneous functions of degree $d_i-d_j-1$. It turns out that any semisimple bi-flat $F$ manifold satisfying a natural additional assumption can be obtained in this way. Finally we show that three dimensional semisimple bi-flat $F$-manifolds are parametrized by solutions of the full family of Painlev\'e VI.