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Local diffeomorphism
In mathematics, a local diffeomorphism is a smooth map f : M → N between smooth manifolds such that for every point p of M there exists an open neighbourhood U of p such that f(U) is open in N and f|U : U → f(U) is a diffeomorphism.
Note that:
- Every local diffeomorphism is also a local homeomorphism and therefore an open map.
- A diffeomorphism is just a bijective local diffeomorphism.
According to the inverse function theorem, a smooth map f : M → N is a local diffeomorphism if and only if the derivative Dfp : TpM → Tf(p)N is a linear isomorphism for all points p in M. Note that this implies that M and N must have the same dimension.
10-26-2009 08:16:03
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The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details