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For a definition of differentiable manifold, we require transition maps to be diffeomorphisms. Do we also require that for every chart $(U,V,\phi)$ map $\phi: U \to V$ is diffeomorphism, or just a homeomorphism?

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You do not require because it wouldn't make sense! But a posteriori, yes they are diffeomorphisms. –  Georges Elencwajg Jun 24 '12 at 12:22
    
I think I am confused. Why it would not make sense to say at each point manifold looks locally as Rn, in a sense map $\phi: U\to V$ is diffeomorphism? Is this stronger definition? –  dmm Jun 24 '12 at 12:28
    
At first the manifold is just a set. The structure is put onto it by the charts. –  Nick Alger Jun 24 '12 at 12:29
    
Yes, thanks for comment. Let me be more explicit. When we want to put differentiable structure on a topological manifold do we only require transition maps to be diffeomorphisms, or maps from charts should be diffeomoprhisms as well? I know that the second condition implies the first one, but I am not sure about the other way around. –  dmm Jun 24 '12 at 12:35
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up vote 3 down vote accepted

Given a topological manifold you only require that the charts are homeomorphisms and that the transition functions are diffeomorphisms. Saying that the charts are differentiable does not make sense in the topological setting. But if you have a differentiable atlas you can define that any map from or to the manifold is differentiable if its concatenations with the charts are differentiable. So after you have defined such a differentiable structure the charts are automatically diffeomorphisms.

As for $dφ_p$: After you have defined such a differentiable structure on the manifold, the map $φ$ is just another map between two manifolds where one happens to be $V ⊂ ℝ^n$. Then the notion of tangent space is defined on both of them and for $φ: U → V$ when $p\in U, φ(p) = 0 \in V$ we have that $dφ_p$ is the induced map $T_pU → T_0V \simeq ℝ^n$. Depending on how you construct the tangent spaces also the understanding of $dφ$ will vary. If, like in your MIT text book, you define $T_pU$ to be the space of curves in $U$ through $p$ that “point in the same direction”, all you have to do, to calculate $dφ_p$ is take a representative curve $γ$ in $U$ and push it to $V$ like $φ∘γ$ which gives you a curve in $V$ that represents some element of the tangent space $T_0V$.

Of course in this special case for $φ$ being a chart you can also give a basis of $T_pU$ in terms of $φ$ just by taking the coordinate curves $γ_i : t ↦ (0, …, 0, t, 0, …, 0) \in V$ and then defining $X_i := [φ^{-1}∘γ_i]_{T_pM}$ where $[\cdot]$ stands for taking all curves in the same direction.

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Thank you. As I understood when we have differential structure then any chart function $\phi$ is diffeomorphism as $\bf{1}_{{Id}}=\phi\circ \phi^{-1}$ is diffeomoprhism. How does one make sense out of $d\phi_{p}$? Why I am asking this is that I am stuck on link diagram 3, Remark 2.14, page 15. Can someone provide me with clarification please? Thank you again! –  dmm Jun 24 '12 at 14:10
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