Integrable Hierarchies and F-Manifolds with Compatible Connection

We study the geometry of integrable systems of hydrodynamic type of the form \(w_t=X\circ w_x\) where \(\circ \) is the product of a regular F-manifold. In the first part of the paper, we present a general construction of a connection compatible with the F-manifold structure starting from a frame of vector fields defining commuting flows of hydrodynamic type. In the second part of the paper, using this construction, we study regular F-manifolds with compatible connection and Euler vector field, \((\nabla ,\circ ,e,E)\), associated with integrable hierarchies obtained from the solutions of the equation \(d\cdot d_L \,a_0=0\) where \(L=E\circ \). In particular, we show that n-dimensional F-manifolds associated to regular operators L are classified by n arbitrary functions of a single variable. Moreover, we show that flat connections \(\nabla \) correspond to linear solutions \(a_0\).