Pullback bundle

Let $\pi:E\to M$ be a smooth fiber bundle and let $f:N\to M$ be a smooth map . The pullback bundle $f^*E\to N$ is defined as the fiber product

Let $\pi_f:f^*E\to N$ be the restriction of the projection $\mathrm{pr}_1:N\times E\to N$. Then $\pi_f$ is a smooth fiber bundle over $N$ with the same typical fiber as $E\to M$.

There is a canonical map $\widetilde f:f^*E\to E$, $\widetilde f(n,e)=e$, which is a bundle morphism covering $f$. Locally, if $E|_U\cong U\times F$, then $(f^*E)|_{f^{-1}(U)}\cong f^{-1}(U)\times F$, so pullback respects local trivializations.

Examples

1. Restriction to a submanifold: if $i:Z\hookrightarrow M$ is an embedding and $E\to M$ is a bundle, then $i^*E\to Z$ is the restricted bundle $E|_Z$.
2. Pullback of the tangent bundle: for $f:N\to M$, the bundle $f^*(TM)\to N$ has fiber $(f^*TM)_n\cong T_{f(n)}M$ and is used to view $df$ as a fiberwise linear map $TN\to f^*TM$.
3. Trivial bundles pull back trivially: if $E=M\times F$, then $f^*E\cong N\times F$.