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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 22nd 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexa minimum, just for completeness <a href="https://ncatlab.org/nlab/revision/conformally+flat+manifold/1">v1</a>, <a href="https://ncatlab.org/nlab/show/conformally+flat+manifold">current</a>

   a minimum, just for completeness

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorGuest
   • CommentTimeApr 11th 2021
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   Author: Guest
   Format: MarkdownItexI believe your definitions are inconsistent: You define [[conformally flat manifold]] by demanding that it can be taken to a flat manifold locally by a [[conformal transformation]]. Then you specify that by "conformal transformation" you really mean a Weyl transformation (locally rescaling the metric). The latter leads to the correct definition. However, on the page for [[conformal transformation]], you define those as diffeomorphisms for which the metric pulls back to the same metric up to a Wel transformation. Such transformations result in a more restrictive class of Weyl transformations. So with your given definition of [[conformally transformation]], it appears wrong to say that a [[conformally flat manifold]] is one that can be locally conformally transformed to flat. -Daniel Ranard

   I believe your definitions are inconsistent:

   You define conformally flat manifold by demanding that it can be taken to a flat manifold locally by a conformal transformation. Then you specify that by “conformal transformation” you really mean a Weyl transformation (locally rescaling the metric). The latter leads to the correct definition. However, on the page for conformal transformation, you define those as diffeomorphisms for which the metric pulls back to the same metric up to a Wel transformation. Such transformations result in a more restrictive class of Weyl transformations. So with your given definition of conformally transformation, it appears wrong to say that a conformally flat manifold is one that can be locally conformally transformed to flat.

   -Daniel Ranard

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeApr 11th 2021
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   Author: Urs
   Format: MarkdownItexThanks for the heads-up. For what it's worth, doesn't it agree exactly with what Wikipedia states, too? Should we sprinkle in some "locally"? I don't quite have the leisure to deal with this right now, maybe later. But if you have the energy, please feel invited to edit as you see the need!

   Thanks for the heads-up. For what it’s worth, doesn’t it agree exactly with what Wikipedia states, too? Should we sprinkle in some “locally”?

   I don’t quite have the leisure to deal with this right now, maybe later. But if you have the energy, please feel invited to edit as you see the need!