## From Darboux-Egorov system to bi-flat \$F\$-manifolds    [PDF]

Alessandro Arsie, Paolo Lorenzoni
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,*,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 Painlev\'e VI equation, admitting among its solutions hypergeometric functions. Finally we comment on some open problems of wide scope related to bi-flat \$F\$-manifolds.
View original: http://arxiv.org/abs/1205.2468