Conformal Killing vector field

<p>In <a href="/facts/Conformal_geometry/tuU181cE">conformal geometry</a>, a conformal Killing vector field  on a <a href="/facts/Manifold/rXPERJtu">manifold</a> of <a href="/facts/Dimension/0ktmUyac">dimension</a> <i>n</i> with  <a href="/facts/Metric_tensor/tbqek1gJ">(pseudo) Riemannian metric</a> 
  
    
      
        g
      
    
    {\displaystyle g}
  
 (also called a conformal Killing vector, CKV, or conformal colineation), is a vector field 
  
    
      
        X
      
    
    {\displaystyle X}
  
 whose (locally defined) <a href="/facts/Flow_(mathematics)/CrmJPfeV">flow</a> defines <a href="/facts/Conformal_transformation/cEpQnMad">conformal transformations</a>, that is, preserve 
  
    
      
        g
      
    
    {\displaystyle g}
  
 up to scale and preserve the conformal structure. Several equivalent formulations, called the conformal Killing equation, exist in terms of the <a href="/facts/Lie_derivative/1KSJvrzk">Lie derivative</a> of the flow e.g. 
  
    
      
        
          
            
              L
            
          
          
            X
          
        
        g
        =
        λ
        g
      
    
    {\displaystyle {\mathcal {L}}_{X}g=\lambda g}
  
 
for some function 
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
 on the manifold. For 
  
    
      
        n
        ≠
        2
      
    
    {\displaystyle n\neq 2}
  
 there are a finite number of solutions, specifying the <a href="/facts/Conformal_symmetry/DlegNm7N">conformal symmetry</a> of that space, but in two dimensions, there is an <a href="/facts/Conformal_field_theory/McVlk4hR">infinity of solutions</a>. The name Killing refers to <a href="/facts/Wilhelm_Killing/W27PB2MJ">Wilhelm Killing</a>, who first investigated <a href="/facts/Killing_vector_field/Ane9xIdV">Killing vector fields</a>.
</p>

Conformal Killing vector field open-in-new

Conformal Killing vector field