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Let $i:N\to M$ be a smooth embedding, $\pi:E\to N$ a vector bundle and $s_0:N\to E$ is its zero section.
I have an open neighborhood $U$ of $s_0(N)$ in $E$, and $f:U\to M$ is a smooth map such that $f\circ s_0=i$ and $T_xf$ is bijective for any $x\in s_0(N)$.
How could I prove that there exists an open neighborhood $V$ of $s_0(N)$ in $U$ such that $f|_V$ is a diffeomorphism onto its image?

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inverse function theorem? – OrbiculaR Feb 13 '11 at 21:26
While I agree that the inverse function theorem will enter in somehow, I'm not sure how you'd guarantee that $s_0(N)\subseteq V$... – Jason DeVito Feb 13 '11 at 22:31
Maybe you could adapt the tubular neighorhood theorem – Sam Feb 14 '11 at 16:51
@S.L.: Thanks, I'll prove to adapt the last three paragraphs in the proof at page 256 of Lee, as you adressed to me. – Giuseppe Tortorella Feb 14 '11 at 18:20

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