  Considering three-dimensional Chern-Simons theory, either coupled to matter
or with a Yang-Mills term, we show the validity of a trace identity, playing
the role of a local form of the Callan-Symanzik equation, in all orders of
perturbation theory. From this we deduce the vanishing of the $\beta$-function
associated to the Chern-Simons coupling constant and the full finiteness in the
case of the Yang-Mills Chern-Simons theory. The main ingredient in the proof of
the latter property is the noninvariance of the Chern-Simons form under the
gauge transformations. Our results hold for the three-dimensional Chern-Simons
model in a general Riemannian manifold.
