Tensor Fields & Differential Forms Part 1

Problem 2.17 Let $(E, \pi, M)$ be a $C^{\infty}$ vector bundle with fibre $\mathbb{F}^{n}$, where $\mathbb{F}=$ $\mathbb{R}, \mathbb{C}$ or $\mathbb{H}$. Prove that the homotheties $$ h: \mathbb{F} \times E \rightarrow E, \quad(\lambda, y) \mapsto h(\lambda, y)=\lambda y $$ are $C^{\infty}$.

Problem 2.18 Show that for a $C^{\infty}$ vector bundle $\xi=(E, \pi, M)$ with fibre $\mathbb{R}^{n}$, triviality is equivalent to the existence of $n C^{\infty}$ global sections, linearly independent at each point.

Problem 2.19 Prove that the infinite Möbius strip $M$ (see Problem 1.31) can be considered as the total space of a vector bundle over $S^{1}$. Specifically: (i) Determine the base space, the fibre and the projection map $\pi$. (ii) Prove that the vector bundle $\left(M, \pi, S^{1}\right)$ is locally trivial but not trivial.