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

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 (2005) [7], 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,∇₁,∇₂,∘,∗,e,E), where (M,∘) is an F-manifold, e is the identity of the product ∘, ∇₁ is a flat connection compatible with ∘ and satisfying ∇₁e=0, while E is an eventual identity giving rise to the dual product ∗, and ∇₂ is a flat connection compatible with ∗ and satisfying ∇₂E=0. Moreover, the two connections ∇₁ and ∇₂ are required to be hydrodynamically almost equivalent in the sense specified by Arsie and Lorenzoni (2012) [6]. First we show, similarly to the way in which Frobenius manifolds are constructed starting from Darboux-Egorov systems, that 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 automatically possesses 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 three bi-flat F-manifolds can be obtained by solutions of a two parameter 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.