# Differential structures on $\mathbb{R}$

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) ?

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 $$\psi^{-1}\circ f \circ \varphi$$.