Differential Geometry of Strongly Integrable Systems of Hydrodynamic Type

Here the matrix (gij) (assumed nondegenerate) defines a pseudo-Riemannian metric (with upper indices) of zero curvature on the u-space, Fjk i = Fjki(u) being the corresponding Levi-Civita connection. Thus, the integrability condition can be formulated in terms of the differential geometry of SHT. For such integrable systems S. P. Tsarev [3] found a generalization (for N _> 3) of the hodograph method which lets one "linearize" the system and thus in some sense "integrate" it. More precisely, let the system (i) in the coordinates u I ..... u N be already diagonal: u~, ~ = t , . . . , N , (4 )