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Kernels, and Representations of Diff. Forms.

  1. Nov 12, 2012 #1


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    Hi, All:

    I need some help with some "technology" on differential forms, please:

    1)Im trying to understand how the hyperplane field Tx[itex]\Sigma[/itex]<
    TpM on M=[itex]\Sigma[/itex] x S1 , where [itex]\Sigma[/itex]

    is a surface, is defined as the kernel of the form dθ (the top form on S1).

    I know that T(x,y)(MxN)≈TxM(+)TyM

    But this seems to bring up issues of the dual Tp*M,

    i.e., the cotangent bundle of M .

    How do we define a linear map on a sum vm+vn , each

    a vector on the tangent spaces of M,N at x,y respectively? I think this may have

    to see with the tensor product, but I'm not sure.

    Thanks .
  2. jcsd
  3. Nov 12, 2012 #2


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    dθ is defined on the product as the pullback by the natural projection M x S -- > S. It's value on v + w is then dθ(w) (for v in TM, w in TS). So it's kernel is TM.

    Does this answer your question?
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