Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor

The integrability of m-component systems of hydrodynamic type, u_t = V(u)u_x, by the generalized hodograph method requiresthe diagonalizability of the m x m matrix V(u). This condition is known to be equivalent to the vanishing of the corresponding Haantjes tensor. We generalise this apporach to hydrodynamic chains - infinite-component systems of hydrodynamic type for which the infinity x infinity matrix V(u) is `sufficiently sparse'. For such systems the Haantjes tensor is well defiined, and the calculation of its components involves finite summations only. We illustrate our approach by classifying broad classes of conservative and Hamiltonian hydrodynamic chains with the zero Haantjes tensor. We prove that the vansihing of the Haantjes tensor is the necessary condition for a hydrodynamic chain to possess an infinity of semi-Hamiltonian hydrodynamic reductions, this providing an easy-to-verify condition for the integrability.