# differentiable manifold definition

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

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.
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