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Differential form

  1. Oct 15, 2007 #1
    Let M be a smooth manifold. Locally we can choose 1-forms [tex]\omega[/tex][tex]^{1}[/tex],[tex]\omega[/tex][tex]^{2}[/tex],...[tex]\omega[/tex][tex]^{n}[/tex] whish span M[tex]^{*}_{q}[/tex] for each q. Then are there vector fields X[tex]_{1}[/tex], X[tex]_{2}[/tex], ...,X[tex]_{n}[/tex] with [tex]\omega[/tex][tex]^{i}[/tex](X[tex]_{j}[/tex])=[tex]\delta^{i}_{j}[/tex]? Here [tex]\delta^{i}_{j}[/tex] is Kronecker delta.
    By vector fields, I meant vector fields on M.
    I think there are such vector fields on small neighborhood B in M.(since M* is locally
    trivial, we can think of M* restricted to B as B X R^n. And we can find such 1-forms w_1, w_2,...w_n which span M* at each p in B. And of course we can find vector fields X[tex]_{1}[/tex], X[tex]_{2}[/tex], ...,X[tex]_{n}[/tex] on B such that
    [tex]\omega[/tex][tex]^{i}[/tex](X[tex]_{j}[/tex])=[tex]\delta^{i}_{j}[/tex].
    But I am wondering if we can extend this vector fields to whole of M.
     
    Last edited: Oct 15, 2007
  2. jcsd
  3. Oct 15, 2007 #2
    You started with 1-forms which were chosen locally. So before answering your latter question, you should think about whether you can choose the 1-forms globally on M.
     
  4. Oct 16, 2007 #3
    I think we can always find globally defined 1-forms w_1, w_2,...w_n on M which in some small neighborhood B, they span M* for each p in B. If not please correct me.
    My question came from the proof of Frobenius integrability theorem in Spivak Volume 1.
    It is a chapter 7 Theorem 14. He starts the proof with locally defined 1-forms w_1,w_2,...,w_n. But in the proof he says:Let X_1, X_2,... X_n be the vecor fields with
    w_i(X_j)= delta^i_j. Here, I think he is referring vector fields on M.
     
  5. Oct 16, 2007 #4
    In general such X does not exist since e.g. on sphere you cannot construct nowhere vanishing vector field. So I think the proof refers to locally defined fields.
     
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