Theorem: Pullback of a principal bundle is a principal bundle

The pullback construction sends principal bundles to principal bundles functorially in the base map.

Theorem: Pullback of a principal bundle is a principal bundle

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

Theorem (pullback principal bundle)

Define the pullback total space

f^*P := \{(n,p)\in N\times P \mid f(n)=\pi(p)\},

with projection $\pi_N:f^*P\to N$ given by $\pi_N(n,p)=n$ . Equip $f^*P$ with the right $G$ -action

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

Then $\pi_N:f^*P\to N$ is a principal $G$ -bundle, called the pullback bundle of $P$ along $f$ .

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

Examples

Pullback of a trivial bundle. If $P\cong M\times G$ , then $f^*P\cong N\times G$ by the evident identification.
Restriction to a submanifold. If $i:Z\hookrightarrow M$ is an embedding, then $i^*P$ is the restriction of $P$ to $Z$ ; its total space is $\pi^{-1}(Z)\subset P$ .
Pullback along a covering map. If $f:N\to M$ is a covering and $P\to M$ is nontrivial, $f^*P\to N$ may become trivial; for circle bundles this reflects the effect of the degree of $f$ on the corresponding cohomology class.