Coframe

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a coframe or coframe field on a <a href="/facts/Smooth_manifold/aSnbXKcV">smooth manifold</a> 
  
    
      
        M
      
    
    {\displaystyle M}
  
 is a system of <a href="/facts/One-form/NQLbWmtD">one-forms</a> or <a href="/facts/Covector/oGh7Asrz">covectors</a> which form a <a href="/facts/Vector_space/M1pxMLx2">basis</a> of the <a href="/facts/Cotangent_bundle/dewBXXxr">cotangent bundle</a> at every point.  In the <a href="/facts/Exterior_algebra/rdN4TvpR">exterior algebra</a> of 
  
    
      
        M
      
    
    {\displaystyle M}
  
, one has a natural map from 
  
    
      
        
          v
          
            k
          
        
        :
        
          ⨁
          
            k
          
        
        
          T
          
            ∗
          
        
        M
        →
        
          ⋀
          
            k
          
        
        
          T
          
            ∗
          
        
        M
      
    
    {\displaystyle v_{k}:\bigoplus ^{k}T^{*}M\to \bigwedge ^{k}T^{*}M}
  
, given by 
  
    
      
        
          v
          
            k
          
        
        :
        (
        
          ρ
          
            1
          
        
        ,
        …
        ,
        
          ρ
          
            k
          
        
        )
        ↦
        
          ρ
          
            1
          
        
        ∧
        …
        ∧
        
          ρ
          
            k
          
        
      
    
    {\displaystyle v_{k}:(\rho _{1},\ldots ,\rho _{k})\mapsto \rho _{1}\wedge \ldots \wedge \rho _{k}}
  
.  If 
  
    
      
        M
      
    
    {\displaystyle M}
  
 is 
  
    
      
        n
      
    
    {\displaystyle n}
  
 dimensional, a coframe is given by a section 
  
    
      
        σ
      
    
    {\displaystyle \sigma }
  
 of 
  
    
      
        
          ⨁
          
            n
          
        
        
          T
          
            ∗
          
        
        M
      
    
    {\displaystyle \bigoplus ^{n}T^{*}M}
  
 such that 
  
    
      
        
          v
          
            n
          
        
        ∘
        σ
        ≠
        0
      
    
    {\displaystyle v_{n}\circ \sigma \neq 0}
  
.  The inverse image under 
  
    
      
        
          v
          
            n
          
        
      
    
    {\displaystyle v_{n}}
  
 of the complement of the zero section of 
  
    
      
        
          ⋀
          
            n
          
        
        
          T
          
            ∗
          
        
        M
      
    
    {\displaystyle \bigwedge ^{n}T^{*}M}
  
 forms a 
  
    
      
        G
        L
        (
        n
        )
      
    
    {\displaystyle GL(n)}
  
 <a href="/facts/Principal_bundle/bbB2D94B">principal bundle</a> over 
  
    
      
        M
      
    
    {\displaystyle M}
  
, which is called the <a href="/facts/Frame_bundle/Gugt3qyL">coframe bundle</a>.
</p>

<ul><li>Manuel Tecchiolli (2019). <a href="https://doi.org/10.3390%2Funiverse5100206">"On the Mathematics of Coframe Formalism and Einstein-Cartan Theory -- A Brief Review"</a>. <i>Universe</i>. 5(10) (Torsion Gravity): 206. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/2008.08314">2008.08314</a>. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2019Univ....5..206T">2019Univ....5..206T</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.3390%2Funiverse5100206">10.3390/universe5100206</a>.</li></ul>

Coframe open-in-new

Coframe