Let $r \in \mathbb N$, let $f: N \to M$ be a smooth map between smooth manifolds, and let $\pi: E \to M$ be a smooth real [vector bundle](/page/Vector%20Bundle) of rank $r$. Define the pullback set $f^*E := \{(n,e) \in N \times E : f(n) = \pi(e)\}$ with projection $\rho: f^*E \to N$ given by $\rho(n,e) = n$. Then $\rho: f^*E \to N$ admits a natural smooth real vector bundle structure of rank $r$, with fiber over each $n \in N$ canonically equal to $E_{f(n)}$.