Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

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
   • PermaLink
   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
   • PermaLink
   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!