  An alternative interpretation of the conformal transformations of the metric
is discussed according to which the latter can be viewed as a mapping among
Riemannian and Weyl-integrable spaces. A novel aspect of the conformal
transformation's issue is then revealed: these transformations relate
complementary geometrical pictures of a same physical reality, so that, the
question about which is the physical conformal frame, does not arise. In
addition, arguments are given which point out that, unless a clear statement of
what is understood by "equivalence of frames" is made, the issue is a semantic
one. For definiteness, an intuitively "natural" statement of conformal
equivalence is given, which is associated with conformal invariance of the
field equations. Under this particular reading, equivalence can take place only
if the metric is defined up to a conformal equivalence class. A concrete
example of a conformal-invariant theory of gravity is then explored. Since
Brans-Dicke theory is not conformally invariant, then the Jordan's and
Einstein's frames of the theory are not equivalent. Otherwise, in view of the
alternative approach proposed here, these frames represent complementary
geometrical descriptions of a same phenomenon. The different points of view
existing in the literature are critically scrutinized on the light of the new
arguments.
