I have an issue with an exercise regarding the construction of the restriction of a given vector bundle.
Here is what I have:
$\pi: E \to M$ is a vector bundle. Then for every $p \in M$ the fiber $E_p:=\pi^{-1}(\{p\})$ comes with the structure of a k-dimensional real vector space.
Moreover, for every $p \in M$ there is an open set $U \subset M$ with $p \in U$ and a diffeomorphism $\phi: \pi^{-1}(U) \to U \times \mathbb{R}^k$ such that
$$ \pi_k \circ \phi=\pi $$
where $\pi_k: U \times \mathbb{R}^k \to U, (p,v) \mapsto p$ and for each $p \in U$ the map $\pi_{\mathbb{R}^k} \circ \phi |_{E_p} \to \mathbb{R}^k$ is a vector space isomorphism.
We can define $E |_N:=\cup_{p \in N}$ and consider the restriction $\phi_N: \pi^{-1} |_N (U \cap N) \to (U \cap N) \times \mathbb{R}^k$. This should induce a vector bundle structure.
But now I'm uncertain about how to define $f$. Since $E |_N \subset E$ I thought about taking the inclusion map from $E |_N$ to $E$. But then I'm not sure how to prove $i \circ \pi_N=\pi \circ f$.