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?
costructing a diffeomorphism
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differential-geometry
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1inverse function theorem? – 2011-02-13
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0While I agree that the inverse function theorem will enter in somehow, I'm not sure how you'd guarantee that $s_0(N)\subseteq V$... – 2011-02-13
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0Maybe you could adapt the [tubular neighorhood theorem](http://books.google.ch/books?id=eqfgZtjQceYC&lpg=PP1&ots=xXb2J9v_B0&dq=introduction%20to%20smooth%20manifolds&pg=PA255#v=onepage&q&f=false) – 2011-02-14
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0@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. – 2011-02-14