demonstrated simply by taking into account the coupling of the scalar field to matter and the varying units in the Einstein frame, but in other cases the physical equivalence is not obvious and it does not seem to hold. At the very least, this equivalence, if it is valid at all, must be defined in precise terms and discussed in ways that are far from obvious. For this reason, it would be too simplistic to dismiss the issue of the conformal frame as a pseudo-problem that has been solved for all physical situations of interest.

Conformally related metrics and Lagrangians
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Taking in mind the above discussion, we want to seek for geometrical structures that are conformally invariant.

Our aim is to compare cosmological models coming from scalar-tensor gravity in order to select conformal quantities in view of a possible physical meaning. This is a very delicate issue that has to be discussed in details. On one hand, an invariant quantity, (i.e. a quantity that remains the same under conformal transformation) should have a physical meaning. However, in cosmology, this statement does not necessarily hold since the problem of equivalence between the two frames is not well posed and it 