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In this paper, we delve into the significance of local scale symmetry and therole of the associated noether current, within gravitational theories which arebased in non-riemannian background space. Our focus is in Weyl and inRiemann-Cartan geometry based gravitational theories. We show that local scalesymmetry is associated with vanishing noether current whenever there are notnew propagating gravitational degrees of freedom beyond the two polarizationsof the massless graviton. In contrast, in local scale invariant theories wherethere are more than two propagating gravitational degrees of freedom, localscale symmetry is associated with nonvanishing noether current. The knownresult that Weyl symmetry has vanishing noether current, is generalized tonon-riemannian gravitational theories. An exception are the local scaleinvariant gravitational theories with vectorial nonmetricity, where theassociated noether current is nonvanishing.

Local scale symmetry in non-riemannian geometry based gravitational theories and the role of the noether current