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In the professor Lee's introduction to smooth manifolds 2nd edition, the notion of diffeomorphism is defined for smooth manifolds with or without boundary. However, I saw some propositions that seems also to be true for smooth manifolds with boundary where professor Lee made the hypothesis only for smooth manifolds.

Example 2.14(b)on page 38: If $M$ is any smooth manifold and $(U,\varphi)$ is a smooth coordinate chart on $M$; then $\varphi:U\to \varphi(U)$ is a diffeomorphism.

Theorem 2.17 (Diffeomorphism Invariance of Dimension) on page 39: A nonempty smooth manifold of dimension $m$ cannot be diffeomorphic to an $n$-dimensional smooth manifold unless $m=n$.

Usually if some proposition is also true for the smooth manifolds with boundary, he will state clearly in the hypothesis. But this time he doesn't, so I wonder if there is any subtle reasons making the two propositions above no-longer true for smooth manifolds with boundary(I feel like a simple modification of the proof works also for the smooth manifolds with boundary), or professor Lee simply forgot to say "smooth manifolds with or without boundary"?

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You are right, it is equally true. Look at the chapter of tangent space, he defines the space tangente for points on boundary and it has the same dimension as $M$. So using that it is a diffeo it would implie that $df$ is a isomorphism, what is possible if, and only if, $m=n$.

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  • $\begingroup$ Do you think that $\varphi:U\to \varphi(U)$ is also a diffeomorphism when $M$ has boundary? Why? $\endgroup$
    – No One
    Commented Mar 26, 2016 at 3:25
  • $\begingroup$ take a chart for $\partial M$ as follows: $(U,\phi)$ chart for $M$. Then $(\partial M \cap U, \phi_{\partial M})$ is a chart for $\partial M.$ Take $p \in \partial M.$ Then $\phi_{\partial M}(U\cap \partial M) = \partial \mathbb{H}^n = \mathbb{R}^{n-1}.$ Using that $\phi : U \to \phi(U)$ is a diffeomorphism we get the result. $\endgroup$ Commented Mar 26, 2016 at 3:34
  • $\begingroup$ Notice that $U\cap \partial M$ is open on $\partial M$. Then $\phi_{\partial M}$ is a homeomorphism. But it is also a diffeomorfism for the same reason of the Lee example... $\endgroup$ Commented Mar 26, 2016 at 3:39
  • $\begingroup$ Thank you and +1, I need to think maybe for a couple of days before I take you solution~~~ $\endgroup$
    – No One
    Commented Mar 26, 2016 at 3:41
  • $\begingroup$ there is no reason to accept my solution now or in any moment. I suggest you to look at the chapters 3 and 5.... $\endgroup$ Commented Mar 26, 2016 at 3:43

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