  The present work deals with two different but subtilely related kinds of
conformal mappings: Weyl rescaling in $d>2$ dimensional spaces and SO(2,d)
transformations. We express how the difference between the two can be
compensated by diffeomorphic transformations. This is well known in the
framework of String Theory but in the particular case of $d=2$ spaces. Indeed,
the Polyakov formalism describes world-sheets in terms of two-dimensional
conformal field theory. On the other hand, B. Zumino had shown that a classical
four-dimensional Weyl-invariant field theory restricted to live in Minkowski
space leads to an SO(2,4)-invariant field theory. We extend Zumino's result to
relate Weyl and SO(2,d) symmetries in arbitrary conformally flat spaces (CFS).
This allows us to assert that a classical $SO(2,d)$-invariant field does not
distinguish, at least locally, between two different $d$-dimensional CFSs.

High Energy Physics - Theory;General Relativity and Quantum Cosmology