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  • 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,

  1. if this $e^{-2\omega(x)}$ is arbitrary differentiable and
  2. 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.

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  • $\begingroup$ In case you are confused about why the linked question is a duplicate: Conformal transformations are a subset of diffeomorphisms, so the reasons why conformal transformations and Weyl transformations are different are precisely the same reasons diffeomorphisms and Weyl transformations are different. $\endgroup$
    – ACuriousMind ♦
    Commented May 6, 2021 at 23:27
  • $\begingroup$ But there is no accepted answer or final correct answer in the post... $\endgroup$
    – ann marie cœur
    Commented May 6, 2021 at 23:42
  • $\begingroup$ Also Conformal transformations though are differentiable, thus only a subset of diffeomorphisms, So diffeomorphism is more general than Conformal transformation... So my question on diffeomorphism may result in different answers!? $\endgroup$
    – ann marie cœur
    Commented May 6, 2021 at 23:43
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    $\begingroup$ 1. There are five answers with positive scores on the other post. "accepted" is just a mark the asker hands out, it is not a sign of definitiveness or correctness, just a sign one specific person found one answer most helpful. 2. If you think the answer to your question is different, you should edit your question and explain why you think that. The only way that could be I could see is if Weyl transformations were diffeomorphisms that just were not conformal, but the answers to the other question explain at length that Weyl transformations are not diffeomorphisms/coordinate changes. $\endgroup$
    – ACuriousMind ♦
    Commented May 6, 2021 at 23:48
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    $\begingroup$ It's just confusing nomenclature because the Weyl transformation act on the metric exactly like conformal transformations. As you can see in the duplicate, it's not uncommon for people to confuse Weyl transformations with conformal transformations. $\endgroup$
    – ACuriousMind ♦
    Commented May 7, 2021 at 0:15

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No to both. A Weyl transformation will send a metric to another metric that will not be the image of the first under diffeomorphism. Some diffeomorphisms have the effect of a Weyl transformation on the metric; those are called conformal maps and are quite special (finite in number outside of dimension 2 and 1).

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  • $\begingroup$ Could you say more for "Some diffeomorphisms have the effect of a Weyl transformation on the metric; those are called conformal maps and are quite special (finite in number outside of dimension 2 and 1)." and give examples and refs? thanks! +1 $\endgroup$
    – ann marie cœur
    Commented May 6, 2021 at 23:50