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Could any one give me hint how to show that the zero section of any smooth vector bundle is smooth?

Zero section is a map $\xi:M\rightarrow E$ defined by $\xi(p)=0\qquad\forall p\in M.$

2 Answers 2

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Smoothness can be checked locally on $M$ and locally $E$ is trivial.

Can you use these two facts to conclude what you want?

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    @user9547, what I mean is that a function is smooth if it is locally smooth.2017-01-15
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Say $E\to M$ is a smooth vector bundle, and let $p\in M$. Then there is a neighborhood $U\subseteq M$ of $p$ and a smooth local trivialization $\Phi:\pi^{-1}(U)\to U\times\mathbb R^k$ of $E$ over $U$. What can you say about $(\Phi\circ\zeta)(q)$ where $q\in U$? Can you conclude that $\Phi\circ\zeta$ is smooth? Now remember that $\Phi$ is a diffeomorphism.

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    @user59083 Hi. I did what you say in your answer, but I get that $\xi$ is smooth in $U$. Maybe this is a silly question but how do I extent this to conclude smoothness in $M$?2017-01-14