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

Abstract

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 $(M,{\nabla }_1,{\nabla }_2,\circ ,\ast ,e,E)$, 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 Painlevé VI equation, admitting among its solutions hypergeometric functions. Finally we comment on some open problems of wide scope related to bi-flat $F$-manifolds.

32 pages, eliminated a remark at the end of proof of Theorem 6.2

References in corpus (3)

• $F$-manifolds with eventual identities, bidifferential calculus and twisted Lenard-Magri chains
• F-manifolds with flat structure and Dubrovin's duality
• Oriented Associativity Equations and Symmetry Consistent Conjugate Curvilinear Coordinate Nets

Cited by in corpus (12)

• Flat Structure on the Space of Isomonodromic Deformations
• Complex reflection groups, logarithmic connections and bi-flat F-manifolds
• Flat F-manifolds, F-CohFTs, and integrable hierarchies
• Darboux-Egorov system, bi-flat $F$-manifolds and Painlevé VI
• A Dubrovin-Frobenius manifold structure of NLS type on the orbit space of $B_n$
• Regular non-semisimple Dubrovin-Frobenius manifolds
• Poisson bracket on 1-forms and evolutionary partial differential equations
• 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
• Frobenius manifolds and quantum groups
• Reciprocal $F$-manifolds