One-form (differential geometry)

In <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a>, a one-form (or covector field) on a <a href="/facts/Differentiable_manifold/aSnbXKcV">differentiable manifold</a> is a <a href="/facts/Differential_form/T9gyHtMv">differential form</a> of degree one, that is, a <a href="/facts/Smooth_function/mffL4ch2">smooth</a> <a href="/facts/Section_(fiber_bundle)/DLnpXS5a">section</a> of the <a href="/facts/Cotangent_bundle/dewBXXxr">cotangent bundle</a>. Equivalently, a one-form on a manifold 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is a smooth mapping of the <a href="/facts/Total_space/SJ7Ek6cI">total space</a> of the <a href="/facts/Tangent_bundle/dq8zW9Kb">tangent bundle</a> of 
 
 
 
 M
 
 
 {\displaystyle M}
 
 to 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
 whose restriction to each fibre is a linear functional on the tangent space. Let 
 
 
 
 ω
 
 
 {\displaystyle \omega }
 
 be a one-form. Then

 
 
 
 
 
 
 
 ω
 :
 U
 
 
 
 →
 
 ⋃
 
 p
 ∈
 U
 
 
 
 T
 
 p
 
 
 ∗
 
 
 (
 
 
 R
 
 
 n
 
 
 )
 
 
 
 
 p
 
 
 
 ↦
 
 ω
 
 p
 
 
 ∈
 
 T
 
 p
 
 
 ∗
 
 
 (
 
 
 R
 
 
 n
 
 
 )
 
 
 
 
 
 
 {\displaystyle {\begin{aligned}\omega :U&\rightarrow \bigcup _{p\in U}T_{p}^{*}(\mathbb {R} ^{n})\\p&\mapsto \omega _{p}\in T_{p}^{*}(\mathbb {R} ^{n})\end{aligned}}}

Often one-forms are described <a href="/facts/Local_property/sdBEwIy2">locally</a>, particularly in <a href="/facts/Local_coordinates/cHmdCEYD">local coordinates</a>. In a local coordinate system, a one-form is a linear combination of the <a href="/facts/Exterior_derivative/p7HA0wze">differentials</a> of the coordinates:

α
          
            x
          
        
        =
        
          f
          
            1
          
        
        (
        x
        )
        
        d
        
          x
          
            1
          
        
        +
        
          f
          
            2
          
        
        (
        x
        )
        
        d
        
          x
          
            2
          
        
        +
        ⋯
        +
        
          f
          
            n
          
        
        (
        x
        )
        
        d
        
          x
          
            n
          
        
        ,
      
    
    {\displaystyle \alpha _{x}=f_{1}(x)\,dx_{1}+f_{2}(x)\,dx_{2}+\cdots +f_{n}(x)\,dx_{n},}

where the 
 
 
 
 
 f
 
 i
 
 
 
 
 {\displaystyle f_{i}}
 
 are smooth functions. From this perspective, a one-form has a <a href="/facts/Covariance_and_contravariance_of_vectors/s3HGMkhz">covariant</a> transformation law on passing from one coordinate system to another. Thus a one-form is an order 1 covariant <a href="/facts/Tensor_field/cAC6F6rF">tensor field</a>.

One-form (differential geometry) open-in-new

One-form (differential geometry)