  Scale-invariant actions in arbitrary dimensions are investigated in curved
space to clarify the relation between scale-, Weyl- and conformal invariance on
the classical level. The global Weyl-group is gauged. Then the class of actions
is determined for which Weyl-gauging may be replaced by a suitable coupling to
the curvature (Ricci gauging). It is shown that this class is exactly the class
of actions which are conformally invariant in flat space. The procedure yields
a simple algebraic criterion for conformal invariance and produces the improved
energy-momentum tensor in conformally invariant theories in a systematic way.
It also provides a simple and fundamental connection between Weyl-anomalies and
central extensions in two dimensions. In particular, the subset of
scale-invariant Lagrangians for fields of arbitrary spin, in any dimension,
which are conformally invariant is given. An example of a quadratic action for
which scale-invariance does not imply conformal invariance is constructed.
