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By definition (http://en.wikipedia.org/wiki/Local_diffeomorphism), that $f$ is a local diffeomorphism of $M$ means for every point $p$ of $M$, there exists a neighborhood $U$ of $p$ such that $f(U)$ is open and $f|_U:U\rightarrow f(U)$ is a diffeomorphism.

However, I think this definition may imply that $f$ is a global diffeomorphism because by definition of diffeomorphism (in Lee's book Manifolds and differential geometry) which is precisely that $f$ is diffeomorphism at every point.

So, do I miss something?

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You miss that a local diffeomorphism needn't to be bijective, for example $\exp(i\cdot) \colon \mathbb R\to S^1$ is a local diffeomorphism. –  martini Oct 22 '12 at 11:26
right....thank you. –  hxhxhx88 Oct 22 '12 at 11:59

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