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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.

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.

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