Coframe

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In mathematics, a coframe or coframe field on a smooth manifold M {\displaystyle M} is a system of one-forms or covectors which form a basis of the cotangent bundle at every point. In the exterior algebra 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)} principal bundle over M {\displaystyle M} , which is called the coframe bundle.

In mathematics, a coframe or coframe field on a smooth manifold

    M
  

{\displaystyle M}

is a system of one-forms or covectors which form a basis of the cotangent bundle at every point. In the exterior algebra 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)}

principal bundle over

    M
  

{\displaystyle M}

, which is called the coframe bundle.

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