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Let $\mathbb{R}$ be the real line with the differential structure from the maximal atlas given by the identity map. And let $\mathbb{R}^{\prime}$ be the real line with the differential structure given by the map $\phi: x \rightarrow x^{\frac{1}{3}}$

(i)- Show that these two differential structures are distinct.

(ii)- Show these smooth manifolds are diffeomorphic. (Hint: try not to use the identity map as a diffeomorphism)

for (i) let $(\mathbb{R},\phi=\operatorname{Id}_\mathbb{R}:\mathbb{R}\to\mathbb{R})$ and $(\mathbb{R},\psi:\mathbb{R}\to\mathbb{R})$ where $\psi(x)=x^{1/3}$. now $(\psi \circ \operatorname{id}_\Bbb R \circ\ \phi^{-1})(x) = x^{1/3}$. This map is not smooth at $0$. Hence the identity map is not a diffeomorphism.

how we can solve (ii) ?

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1 Answer 1

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For (i) you are right: these two charts are not compatible, hence belong to two distinct differential structure.

For (ii), just notice that the map \begin{align} f : \mathbb{R} & \to \mathbb{R}' \\ x & \mapsto x^3 \end{align} is a diffeomorphism. To see this, consider $\varphi$ the identity chart on $\mathbb{R}$ and $\psi$ the cube map on $\mathbb{R}'$, and look at $\psi^{-1}\circ f \circ \varphi$.

This shows that even if these two differentiable structures are not compatible, they are diffeomorphic.

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