Local diffeomorphism Totally Explained

Everything about Local Diffeomorphism totally explained

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 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 Df_p : T_pM → T_f(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.

Local flow diffeomorphisms

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