From the Darboux–Egorov system to bi-flat FF-manifolds

Motivated by the theory of integrable PDEs of hydrodynamic type and by the generalization of Dubrovin’s duality in the framework of FF-manifolds due to Manin (2005) [7], we consider a special class of FF-manifolds, called bi-flat FF-manifolds.A bi-flat FF-manifold is given by the following data (M,∇1,∇2,∘,∗,e,E)(M,∇1,∇2,∘,∗,e,E), where (M,∘)(M,∘) is an FF-manifold, ee is the identity of the product ∘∘, ∇1∇1 is a flat connection compatible with ∘∘ and satisfying ∇1e=0∇1e=0, while EE is an eventual identity giving rise to the dual product ∗∗, and ∇2∇2 is a flat connection compatible with ∗∗ and satisfying ∇2E=0∇2E=0. Moreover, the two connections ∇1∇1 and ∇2∇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 FF-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 FF-manifold, we show that the latter is a strictly larger class.In particular we study in some detail bi-flat FF-manifolds in dimensions n=2,3n=2,3. For instance, we show that in dimension three bi-flat FF-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 FF-manifolds.