Non-Smooth Vector Bundle

Question

As I understand it the requirements for a vector bundle $p:E \to M$ to be a smooth vector bundle is for $p$ to be a smooth map and for the local trivialization maps $p^{-1}(U) \to U \times \mathbb{R}^k$ to be diffeomorphisms. What is an example where $p:E \to M$ is a vector bundle in the continuous sense, $p$ is still a smooth map, but the trivializations fail to be smooth?

Answer 1

Let $\tilde{\mathbb{R}}$ denote $\mathbb{R}$ with its standard topology and a smooth structure given by the chart $\phi:\tilde{\mathbb R}\to\mathbb R$, $\phi(x)=x^3$. Note that $\operatorname{id}_{\mathbb R}:\tilde{\mathbb R}\to\mathbb R$ is continuous but not smooth. Then the trivial bundle $p:\tilde{\mathbb R}^2\to\tilde{\mathbb R}$, $p(x,y)=x$ is an example.