Line element in quantum gravity: The example of dsr and noncommutative geometry

We question the notion of line element in some quantum spaces that are expected to play a role in quantum gravity, namely noncommutative deformations of Minkowski spaces. We recall how the implementation of the Leibniz rule for bids to see some of the infinitesimal deformed Poincare transformations as good candidates for Noether symmetries. Then were call the more fundamental view on the line element proposed in noncommutative geometry,and re-interprete at this light some previous results on Connes' distance formula.