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I start to think of this question when I attempt Ex 1.3.9 on Guillemin and Pollack's Differential Topology GP 1.3.9(b) Every manifold is locally expressible as a graph..

I am under the impression that this is true. For the proof of the following question, I need the fact that both the projection function and its inverse are smooth. Is this true?

Let $x_1, \dots, x_N$ be the standard coordinate functions on $\mathbb{R}^N$, and let $X$ be a $k$-dimensional submanifold of $\mathbb{R}^N$. Prove that every point $x \in X$ has a neighborhood on which the restrictions of some $k$-coordinate functions $x_{i_1}, \dots, x_{i_k}$ form a local coordinate system.

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    $\begingroup$ What projection do you mean? $\endgroup$ – Mariano Suárez-Álvarez Jul 3 '13 at 5:56
  • $\begingroup$ Hi @MarianoSuárez-Alvarez thanks. this kind: $$d \varphi: T_x(X) \mapsto \text{ span}(e_{i_1}, \dots, e_{i_k})$$ $\endgroup$ – WishingFish Jul 3 '13 at 15:53
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The projection is smooth. Note that projection is just the restriction of a linear map to the manifold. Since linear functions are smooth, this actually proves what you want (in the context of G&P).

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