Weyl transformation is a local rescaling of the metric tensor $$ g_{ab}\rightarrow e^{-2\omega(x)}g_{ab} $$
Diffeomorphism maps to a theory under arbitrary differentiable coordinate transformations (Diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are smooth.)
Question 1: Is Weyl transformation part of diffeomorphism?
It seems that the answer would be yes,
- if this $e^{-2\omega(x)}$ is arbitrary differentiable and
- if the starting manifold with a $g_{ab}$ is differentiable.
Question 2: Because the gravitational anomaly is also known as diffeomorphism anomaly, related to the diffeomorphism of manifold. Is this correct to say that the gravitational anomaly capture also the anomaly due to Weyl transformation?
p.s. I asked more additional details in a previous post Weyl transformation vs diffeomorphism; conformal invariant vs general in/covariant, but I got no answer. So let us zoom into a specific case. I hope someone can give a definite correct answer this time.