Question: Let M be a surface of class $C^k$ and dimension $m\in \mathbb{N}$ and $M\subset \mathbb{R}^n$ . Consider the following subset $$TM=\{(x,v)\in \mathbb{R}^n\times \mathbb{R}^n; x \in M, v \in T_{x}M\}$$ This subset is called tangent bundle of $M$.
- Show that $TM$ is a surface of class $C^{k-1}$ and dimension $2m$ in $\mathbb{R}^n \times \mathbb{R}^n$.
- If $N$ is another surface and $f: M \rightarrow N$ is a function of class $C^k$, define a tangent function $T_{f}:TM \rightarrow TN$ given by $T_{f}(x,v)=(f(x), f'(x).v)$ and show that $T_{f}$ is $C^{k-1}$ class.
My idea:
To show that $TM$ is a surface I need to find a parametrization (or more than one parametrizations) that is defined in an open set, and such parametrizations cover the subset $TM$ completely. My question is which parametrizatio(s) would I consider?
Here, I just would ask you for some hints.