Conformally flat manifold

<h2 id="examples">Examples</h2>
<ul><li>Every manifold with <a href="/facts/Constant_curvature/VuJg6YlV">constant</a> <a href="/facts/Sectional_curvature/GpI9zQDh">sectional curvature</a> is conformally flat.</li>
<li>Every 2-dimensional pseudo-Riemannian manifold is conformally flat.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>
<ul><li>The <a href="/facts/Line_element/asquI22v">line element</a> of the two dimensional spherical coordinates, like the one used in the <a href="/facts/Geographic_coordinate_system/RkjfcY6k">geographic coordinate system</a>,</li></ul>

d
 
 s
 
 2
 
 
 =
 d
 
 θ
 
 2
 
 
 +
 
 sin
 
 2
 
 
 ⁡
 θ
 
 d
 
 ϕ
 
 2
 
 
 
 
 
 {\displaystyle ds^{2}=d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}\,}
 
,<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> has metric tensor 
 
 
 
 
 g
 
 i
 k
 
 
 =
 
 
 [
 
 
 
 1
 
 
 0
 
 
 
 
 0
 
 
 s
 i
 
 n
 
 2
 
 
 θ
 
 
 
 ]
 
 
 
 
 {\displaystyle g_{ik}={\begin{bmatrix}1&0\\0&sin^{2}\theta \end{bmatrix}}}
 
 and is not flat but with the <a href="/facts/Stereographic_projection/oVyMrWqR">stereographic projection</a> can be mapped to a flat space using the conformal factor 
 
 
 
 
 2
 
 
 (
 1
 +
 
 r
 
 2
 
 
 )
 
 
 
 {\displaystyle 2 \over (1+r^{2})}
 
, where 
 
 
 
 r
 
 
 {\displaystyle r}
 
 is the distance from the origin of the flat space,<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> obtaining

d
 
 s
 
 2
 
 
 =
 d
 
 θ
 
 2
 
 
 +
 
 sin
 
 2
 
 
 ⁡
 θ
 
 d
 
 ϕ
 
 2
 
 
 
 =
 
 
 4
 
 (
 1
 +
 
 r
 
 2
 
 
 
 )
 
 2
 
 
 
 
 
 (
 d
 
 x
 
 2
 
 
 +
 d
 
 y
 
 2
 
 
 )
 
 
 {\displaystyle ds^{2}=d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}\,={\frac {4}{(1+r^{2})^{2}}}(dx^{2}+dy^{2})}
 
.</li>
<li>A 3-dimensional pseudo-Riemannian manifold is conformally flat if and only if the <a href="/facts/Cotton_tensor/NpsP7YzX">Cotton tensor</a> vanishes.</li>
<li>An n-dimensional pseudo-Riemannian manifold for n ≥ 4 is conformally flat if and only if the <a href="/facts/Weyl_tensor/lrXNLWHC">Weyl tensor</a> vanishes.</li>
<li>Every <a href="/facts/Compact_space/d0cgXJH7">compact</a>, <a href="/facts/Simply_connected/xFlta63J">simply connected</a>, conformally Euclidean Riemannian manifold is conformally equivalent to the <a href="/facts/N-sphere/bFu2S7E2">round sphere</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a></li></ul>
<ul><li>The <a href="/facts/Stereographic_projection/oVyMrWqR">stereographic projection</a> provides a coordinate system for the sphere in which conformal flatness is explicit, as the metric is proportional to the flat one.</li></ul>
<ul><li>In <a href="/facts/General_relativity/jVsoIg8X">general relativity</a> conformally flat manifolds can often be used, for example to describe <a href="/facts/Friedmann%25E2%2580%2593Lema%25C3%25AEtre%25E2%2580%2593Robertson%25E2%2580%2593Walker_metric/b6cdgxyF">Friedmann–Lemaître–Robertson–Walker metric</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> However it was also shown that there are no conformally flat slices of the <a href="/facts/Kerr_spacetime/SjL2K6R0">Kerr spacetime</a>.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a></li></ul>
For example, the <a href="/facts/Kruskal%25E2%2580%2593Szekeres_coordinates/VXlGQzbW">Kruskal-Szekeres coordinates</a> have line element

d
        
          s
          
            2
          
        
        =
        
          (
          
            1
            −
            
              
                
                  2
                  G
                  M
                
                r
              
            
          
          )
        
        d
        v
        
        d
        u
      
    
    {\displaystyle ds^{2}=\left(1-{\frac {2GM}{r}}\right)dv\,du}
  
  with metric tensor 
  
    
      
        
          g
          
            i
            k
          
        
        =
        
          
            [
            
              
                
                  0
                
                
                  1
                  −
                  
                    
                      
                        2
                        G
                        M
                      
                      r
                    
                  
                
              
              
                
                  1
                  −
                  
                    
                      
                        2
                        G
                        M
                      
                      r
                    
                  
                
                
                  0
                
              
            
            ]
          
        
      
    
    {\displaystyle g_{ik}={\begin{bmatrix}0&1-{\frac {2GM}{r}}\\1-{\frac {2GM}{r}}&0\end{bmatrix}}}
  
 and so is not flat. But with the transformations 
  
    
      
        t
        =
        (
        v
        +
        u
        )
        
          /
        
        2
      
    
    {\displaystyle t=(v+u)/2}
  
 and 
  
    
      
        x
        =
        (
        v
        −
        u
        )
        
          /
        
        2
      
    
    {\displaystyle x=(v-u)/2}

becomes

d
 
 s
 
 2
 
 
 =
 
 (
 
 1
 −
 
 
 
 2
 G
 M
 
 r
 
 
 
 )
 
 (
 d
 
 t
 
 2
 
 
 −
 d
 
 x
 
 2
 
 
 )
 
 
 {\displaystyle ds^{2}=\left(1-{\frac {2GM}{r}}\right)(dt^{2}-dx^{2})}
 
 with metric tensor 
 
 
 
 
 g
 
 i
 k
 
 
 =
 
 
 [
 
 
 
 1
 −
 
 
 
 2
 G
 M
 
 r
 
 
 
 
 0
 
 
 
 
 0
 
 
 −
 1
 +
 
 
 
 2
 G
 M
 
 r
 
 
 
 
 
 ]
 
 
 
 
 {\displaystyle g_{ik}={\begin{bmatrix}1-{\frac {2GM}{r}}&0\\0&-1+{\frac {2GM}{r}}\end{bmatrix}}}
 
,
which is the flat metric times the conformal factor 
 
 
 
 1
 −
 
 
 
 2
 G
 M
 
 r
 
 
 
 
 {\displaystyle 1-{\frac {2GM}{r}}}
 
.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Weyl%25E2%2580%2593Schouten_theorem/IUhU2hCt">Weyl–Schouten theorem</a></li>
<li><a href="/facts/Conformal_geometry/tuU181cE">conformal geometry</a></li>
<li><a href="/facts/Yamabe_problem/TekKSICb">Yamabe problem</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Ray D'Inverno. "6.13 The Weyl tensor". Introducing Einstein's Relativity. pp. 88–89. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Ray D'Inverno. "6.13 The Weyl tensor". Introducing Einstein's Relativity. pp. 88–89. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Spherical coordinate system - Integration and differentiation in spherical coordinates <a href="/wiki/Spherical_coordinate_system#Integration_and_differentiation_in_spherical_coordinates" target="_blank">/wiki/Spherical_coordinate_system#Integration_and_differentiation_in_spherical_coordinates</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Stereographic projection - Properties. The Riemann's formula <a href="/wiki/Stereographic_projection#Properties" target="_blank">/wiki/Stereographic_projection#Properties</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Kuiper, N. H. (1949). "On conformally flat spaces in the large". Annals of Mathematics. 50 (4): 916–924. doi:10.2307/1969587. JSTOR 1969587. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Garecki, Janusz (2008). "On Energy of the Friedman Universes in Conformally Flat Coordinates". Acta Physica Polonica B. 39 (4): 781–797. arXiv:0708.2783. Bibcode:2008AcPPB..39..781G. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Garat, Alcides; Price, Richard H. (2000-05-18). "Nonexistence of conformally flat slices of the Kerr spacetime". Physical Review D. 61 (12): 124011. arXiv:gr-qc/0002013. Bibcode:2000PhRvD..61l4011G. doi:10.1103/PhysRevD.61.124011. ISSN 0556-2821. S2CID 119452751. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Ray D'Inverno. "17.2 The Kruskal solution". Introducing Einstein's Relativity. pp. 230–231. <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
</ol>

Conformally flat manifold open-in-new

Conformally flat manifold