  Once we put a quantum field theory on a curved manifold, it is natural to
further assume that coupling constants are position dependent. The position
dependent coupling constants then provide an extra contribution to the Weyl
anomaly so that we may attempt to cancel the entire Weyl anomaly on the curved
manifold. We show that such a cancellation is possible for constant Weyl
transformation or infinitesimal but generic Weyl transformation in two and four
dimensional conformal field theories with exactly marginal deformations. When
the Weyl scaling factor is annihilated by conformal powers of Laplacian (e.g.
by Fradkin-Tseytlin-Riegert-Paneitz operator in four dimensions), the
cancellation persists even at the finite order thanks to a nice mathematical
property of the $Q$-curvature under the Weyl transformation.
