Darboux-Egorov system, bi-flat $F$-manifolds and Painlevé VI

Abstract

This is a generalization of the procedure presented in [3] to construct semisimple bi-flat $F$-manifolds $(M,{\nabla }^{(1)},{\nabla }^{(2)},\circ ,\ast ,e,E)$ starting from homogeneous solutions of degree -1 of Darboux-Egorov-system. The Lamé 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 Painlevé VI.

20 pages

References in corpus (4)

• From Darboux-Egorov system to bi-flat $F$-manifolds
• $F$-manifolds with eventual identities, bidifferential calculus and twisted Lenard-Magri chains
• Poisson bracket on 1-forms and evolutionary partial differential equations
• Reciprocal $F$-manifolds

Cited by in corpus (8)

• Complex reflection groups, logarithmic connections and bi-flat F-manifolds
• Flat F-manifolds, F-CohFTs, and integrable hierarchies
• Regular non-semisimple Dubrovin-Frobenius manifolds
• Bi-Hamiltonian structure of the Oriented Associativity Equation
• Integrable systems, Nijenhuis geometry and Lauricella bi-flat structures
• 3-dimensional F-manifolds
• Riemann-Hilbert-Birkhoff inverse problem for semisimple flat $F$-manifolds, and convergence of oriented associativity potentials
• On the universality of integrable deformations of solutions of degenerate Riemann-Hilbert-Birkhoff problems