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Hello I have a very simple question of a proof in the book of Docarmo Differential Geometry. In some page of the book , they said that using the proposition 1 , we can deduce that the parametrizations are also diffeomorphism. How can I prove that? I only know that $ x^{-1} y $ it´s diffeomorphism for all parametrizations y. Using this how can I prove that the function $ x^{-1} $ it´s also differentiable? Here are the definitions used in Docarmo of parametrization, and the proposition 1.Parametrization Definition used on Docarmo

Proposition 1 and the Remark

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The inverse function theorem in tandem with the regularity condition tells you that your paramaterization is a local diffeomorphism, and every bijective local diffeomorphism is a diffeomorphism. This is because being a diffeomorphism is a purely local property, and so locally diffeomorphisms and diffeomorphisms are the same thing (obviously), the only difference is the global property of bijectivity.

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