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This is a question from my differential geometry assignment:

Let $\pi:M\to N$ be a submersion between two smooth manifolds and $X\in \Gamma(TN)$ is a vector field. We need to show that there is a smooth vector field on $M$ that is $\pi$-related to $X$. Also determine a necessary and sufficient condition on $\pi$ for a lift of any vector field to be unique.

So what I've got so far is that, since $\pi$ is a submersion, at each $p\in M$, $\pi_{*p}$ is surjective. So for every $X_{\pi(p)}$ I can choose a $Y_p\in T_pM$ such that $\pi_{*p}Y_p=X_{\pi(p)}$ and define a local smooth vector field in a neighbourhood $U_p$ of $p$. Do it for every $p\in M$ and $\{U_p\}$ gives an open cover of $M$.

My problem here is that, firstly I don't know how I could get a local smooth vector field from $Y_p$. I thought of using an integral curve and extend $Y_p$ according to the curve, but it does not give me a vector field on an open set.

Secondly, in order to get a global smooth vector field, in the overlapping, say a point $r\in U_p\cap U_q$, I am going to have $Y^1_r, Y^2_r$ that come from the two local vector fields defined by two points $p$ and $q$. Then I should probably sum up all the vectors in $T_rM$, then use the partition of unity subordinate to $\{U_p\}$. But I'm not sure how to apply the partition of unity.

I'm stuck now, how can I go from here? Please do not post complete solutions.

Thanks.

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You write $X\in\Gamma(TM)$ but perhaps you mean $X\in\Gamma(TN)$ –  Giuseppe Tortorella May 10 '12 at 11:19
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@Ziping: I think the problem is that your local liftings are not good enough. What you need to do is find an open cover $\{ V_\alpha \}$ of $N$ so that on each $U_\alpha = \pi^{-1} V_\alpha$, there is a lift of $X |_{V_\alpha}$ to $U_\alpha$. This is the hard part. After that, the partition of unity argument should go through. –  Zhen Lin May 10 '12 at 11:26
    
@Giuseppe Oops! Yep. Corrected; Thanks! –  Ziping R May 10 '12 at 11:28
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Your idea is right, define a lift locally and sum those local lifts weighed by a partition of unity. You need only be more careful when defining those local lifts: taking any vector $Y_p$ that projects to $X_{\pi(p)}$ will not result in a smooth local lift. Do you have the theorem that tells you the local form of submersions (and that of immersions)? It is usually derived as a corollary to the local inversion theorem. If you don't, you can also do it by hand with linear algebra. –  Olivier Bégassat May 10 '12 at 11:30
    
@ZhenLin Do you mean I take an open cover of $N$ defined by $V_\alpha=\pi(U_\alpha)$? I can't see how this gives me a local lifting.. –  Ziping R May 10 '12 at 11:34

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