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Definition of a one-form

  1. Nov 8, 2008 #1
    A one-form is something of the form

    [tex]\omega=\omega_\mu dx^\mu[/tex]

    But is it necessary that the components [tex]\omega_\mu[/tex] be components of a type (0,1) tensor?

    For instance, the connection one-form is defined to be

    [tex]{\omega^{\alpha}}_\beta = {\Gamma^\alpha}_{\gamma\beta} \hat{\theta}^\gamma[/tex]

    where [tex]\hat{\theta}^\gamma[/tex] is a basis of the dual tangent space, though not necessarily a coordinate basis. Here the components [tex]{\Gamma^\alpha}_{\gamma\beta} [/tex]--the connection coefficients, i.e., Christoffel symbols-- not only are not those of a type (0,1) tensor, they are not even those of a tensor.

    So is this legitimately a one-form?
     
    Last edited: Nov 8, 2008
  2. jcsd
  3. Nov 8, 2008 #2

    Hurkyl

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    I think with any vector bundle E over your manifold M, it's fair to call any (bundle) mapping TM-->E a "one-form". (i.e. a function that takes tangent vectors and maps them to E-vectors)

    For the usual one-forms, E is just the trivial line bundle MxR-->M -- that is, the one in which scalar fields live. Each (usual) one-form takes a tangent vector field on M and maps it to a scalar field on M. T*M is the bundle in which all such one-forms live.
     
  4. Nov 8, 2008 #3
    Thank you, Hurkyl. I'm getting there, I really am. But I am in chap 7 of Nakahara's Geometry, Topology, and Physics and fibre bundles and such are in chap 9. I think you are saying that the answer to "is it necessary that the components [tex]\omega_\mu[/tex] be components of a type (0,1) tensor? " is "No". Is that right?

    Then again, maybe this means "Yes, it is necessary"? The elements of T*M are identical with the (0,1) tensors, aren't they?
     
  5. Nov 10, 2008 #4
    I heared "connection one-form" but i see 2 componants "two indices" !!!! [tex]{{\omega^{\alpha}}_\beta}'s[/tex] are not one-forms and are not expressed in the natural basis (dx) they are the componants of a 1-1 tensor and a form has never been a 1-1 tensor ! I think :biggrin:
     
  6. Nov 10, 2008 #5
    I think maybe it is just sloppy use of the term "one-form". Though I think it is standard. The wikipedia entry just calls it "connection form" instead of "connection one form" But it is linear in the one-form basis elements [tex]\hat{\theta}^\alpha[/tex]. It is not a k-form where k>1.

    On the other hand, maybe any linear combination of the [tex]\hat{\theta}^\alpha[/tex] is a one form, even though the coefficients are the not components of a (0,1) tensor. I'm not able to glean how strict the definition of a one-form is from the sources I have checked.
     
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