Construction: pullback principal bundle

Let $\pi:P\to M$ be a principal G-bundle with right action, and let $f:N\to M$ be a smooth map between smooth manifolds.

Construction (Pullback bundle)

Define the fiber product

f^*P \;:=\;\{(y,p)\in N\times P:\ f(y)=\pi(p)\}.

Let $\pi_N:f^*P\to N$ be projection to the first factor, $\pi_N(y,p)=y$ . Define a right $G$ -action on $f^*P$ by

(y,p)\cdot g := (y,\ p\cdot g).

Then:

$f^*P$ is a smooth manifold and $\pi_N:f^*P\to N$ is a smooth submersion.
$(f^*P,\pi_N)$ is a principal $G$ -bundle over $N$ .
There is a canonical $G$ -equivariant map $\widetilde f:f^*P\to P$ given by $\widetilde f(y,p)=p$ covering $f$ (i.e. $\pi\circ \widetilde f = f\circ \pi_N$ ).

This construction is functorial: if $g:K\to N$ is another smooth map, then $(f\circ g)^*P$ is canonically isomorphic to $g^*(f^*P)$ as principal bundles.

Examples

Restriction to a submanifold. If $i:U\hookrightarrow M$ is the inclusion of an open set (or an embedded submanifold), then $i^*P\to U$ is the restriction of $P$ to $U$ .

Pullback of a trivial bundle. After choosing a trivialization, the pullback identifies with $N\times G$ ; the identification depends on that choice.

Pullback of local trivializations. If $P$ is described by transition functions $g_{ij}$ on a cover of $M$ , then $f^*P$ is described by the pulled-back transition functions $g_{ij}\circ f$ on the induced cover $\{f^{-1}(U_i)\}$ of $N$ .