Take bundle charts $\Phi_i:E|_{U_i}\to U_i\times\mathbb R^r$ with transition functions $g_{ij}:U_i\cap U_j\to GL(r,\mathbb R)$. Over $f^{-1}(U_i)$ define
$$
\widetilde\Phi_i(p,v)=(p,\operatorname{pr}_2\Phi_i(v)).
$$
Their transition functions are $g_{ij}\circ f$, so they give the fibre product
$$
f^*E=\coprod_{p\in M}E_{f(p)}
$$
a smooth manifold and make its projection to $M$ a rank-$r$ <pullback vector bundle>.

A local section $s:U\to E$ determines the section
$$
p\longmapsto(p,s(f(p)))
$$
of $f^*E$ over $f^{-1}(U)$. It is unique with second component $s\circ f$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 115 / 3 / c / Solution

{title2=$f^*E$}
{wiki}

For a smooth map $f:M\to B$ and a vector bundle $\pi:E\to B$, the pullback bundle is
$$
f^*E=\{(p,v)\in M\times E:f(p)=\pi(v)\}.
$$
Pulling back local trivializations of $E$ makes it a vector bundle over $M$ whose transition functions are the original transition functions composed with $f$.


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 Incoming links: Pullback vector bundle