Differential Forms on Smooth Manifolds

[daishin]

Let M be a smooth manifold. Locally we can choose 1-forms $$\omega$$$$^{1}$$,$$\omega$$$$^{2}$$,...$$\omega$$$$^{n}$$ whish span M$$^{*}_{q}$$ for each q. Then are there vector fields X$$_{1}$$, X$$_{2}$$, ...,X$$_{n}$$ with $$\omega$$$$^{i}$$(X$$_{j}$$)=$$\delta^{i}_{j}$$? Here $$\delta^{i}_{j}$$ 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$$_{1}$$, X$$_{2}$$, ...,X$$_{n}$$ on B such that
$$\omega$$$$^{i}$$(X$$_{j}$$)=$$\delta^{i}_{j}$$.
But I am wondering if we can extend this vector fields to whole of M.

 

[timur]

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.

 

[daishin]

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.

 

[timur]

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.