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Taking only the characteristics as absolute, in the spirit of Arnold's"Geometrical Methods in the Theory of Ordinary Differential Equations"(Springer, 1988), we give an independent of coordinates formulation of generalvariational entropy inequalities for quasilinear equations of first order, thatlocally read as Kruzhkov inequalities, in terms of certain "entropy densities",and in the case of the equation of 2D flat projective structure we get theexpression of the general entropy density from its abstract Rankine-Hugoniotrule for shocks using the projective geometry of the plane.

Entropy conditions for quasilinear first order equations on nonlinear  fiber bundles with special emphasis on the equation of 2D flat projective  structure. I