Problem
Problem 1.59 Let $\mathbb{R}_{\mathrm{id}}$ and $\mathbb{R}_{\varphi}$ be the $C^{\infty}$ manifolds defined, respectively, by the differentiable structures obtained from the atlases $\{(\mathbb{R}, \mathrm{id})\}$ and $\{(\mathbb{R}, \varphi)\}$ on $\mathbb{R}$, where $\varphi: \mathbb{R} \rightarrow \mathbb{R}, \varphi(t)=t^{3}$. Prove that $\mathbb{R}_{\mathrm{id}}$ and $\mathbb{R}_{\varphi}$ are diffeomorphic (see Problem 1.22).