Scale invariance: fake appearances

In this paper we shall show that, unless the affine geometrical structure of the underlying spacetime manifold is specified, there is an ambiguity in the understanding of the scale invariance -- also Weyl invariance -- of the given theory of gravity. In this regard we cast doubts on several well-known theories which are claimed to be scale-invariant setups. We shall show that in a consistent scale invariant theory not only the action but also the geometrical laws which govern the affine structure of the underlying spacetimes are invariant under the Weyl gauge transformations.